Chapter 7 Answers

7.1 Calculating Integrals 1. x 3 + x 2 - 1/2x2 + C 3. eX + x 2 + C 5. -(eos2x)/2 + 3x 2/2 + C 7. -e - x +2sinx+Sx3/3+C 9. 1084/9 tt. lOS /2

13. 844/S 15. 1/12 17. ° 19. 6 21. 3w/4 23. w/12 25. I 27. (e6 - e3)/3 + 3(25/ 3 - I)/S 29. InS 31. 41n2 + 61/24 33.400 35. 116/15 37. (b) e(e2) - e + 3 39. (a) 11

(b) -8 (e) N ote that f~ f(t) dt is negative

41. -2tJe 12 + sinSt4

43. 3 45. (a) °

(b) 5/6

{ -eosx (e) eos(2)-2eosx

y

1.584

0.416

- I

47. 2 + tan- 12 - ilnS 49. 16.4 51. (1/2)(e2 - 1) 53. 16/3 - w

55. y

- I

57. w/4

if 0 < x .;; 2 if 2 < x';; w

x

59. (a) Differentiate the right-hand side. (b) Integrate both sides of the identity. (c) 1/8

61. Use the fact that lan -la and tan -Ib lie in the interval ( - w /2, w /2)

63. 16,000,014 meters

Chapter 7 Answers A.43

65. (a) Evaluate the integral. (b) A = $45,231.46

67. (a) R(t) = 2000e l / 2 - 2000, C(t) = lO00t - (2

(b) $S7,279.90 69. 1 + In(2) - In(l + e) ~ 0.380

7.2 Integration by Substitution 1. Hx 2 + 4)5/ 2 + C

3. -1/4(y8+4y-I)+C 5. - 1/2 tan20 + C 7. sin(x2 + 2x)/2 + C 9. (x 4 + 2)1 / 2/2 + C

tt. -3(t4/ 3 + 1)-1/2/2+ C 13. -cos4(r2)/4 + C 15. tan - l(x4)/4+ C 17. -cos(O + 4) + C 19. (x 5 + x)lol/101 + C

21. ';t2 + 2t + 3 + C 23. (t 2 + 1)3/ 2/3 + C 25. sin 0 - sin30/3 + C 27. Inllnxl + C 29. 2sin - l(x/2) + x~ /2 + C 31. In(l + sin 0) + C 33. - cos(ln t) + C 35. -3(3+ l/x)4/ 3/4+ C 37. (sin 2x)/2 + C 39. m a non-negative integer and n an odd positive

integer, or n a non-negative integer and m an odd positive integer.

7.3 Changing Variables in the Definite Integral

1. 2(313 - 1)/3 5. 2[(25)9/4 - (9)9/4]/9 9. (e - 1)/2

13. ° 17. In( y1 cos( w /8» 21. 4 - tan- I (3) + w/4 23. (a) w /2

(b) w/4 (c) w /8

3. (5/5 - 1)/3 7. 1/7

tt. -1/3 15. 1 19. 1/2

25. The substitution is not helpful in evaluating the integral.

27. (y1 /2)[tan -12y1 - tan -I( y1 /2)]

29. (1/13)ln[(4 + 3y1)/(1 +13)] 31. Let u = x - t.

33. (Sy1 - 2/5)/10

35. (w/27)(I45[i45 - IOfiO) 37. (a) 1/3

(b) Yes.

A.44 Chapter 7 Answers

7.4 Integration By Parts 1. (x + I)sinx + cosx + C 3. x sin 5x/5 + cos5x/25 + C 5. (x 2 - 2)sinx + 2xcosx + C 7. (x + l)e X + C 9. x In(lOx) - x + C

11. (x 3/9)(3Inx - I) + C 13. e3s (9s 2 - 6s + 2) /27 + C 15. (x3 - 4)1/3(x3 + 12)/4 + C 17. t2sin t2 + cos t2 + C 19. -(I/ x)sin(1/ x) - cos(1/ x) + C 21. -[ln(cosx)f /2 + C

23. xcos-I(2x) -~ /2 + C

25. yb/y - I - tan-1b/y - I + C

27. sin2x/2 + C 29. Tbe integral becomes more complicated. 31. (16 + '17)/5 33. 3(3 In 3 - 2)

35. y2[('17/4)2+3'17/4-2]/2-1.

37 . .f3 /8 - '17/24 39. e - 2 41. _(e2". - e- 2")/4 43. H22/3(22/3 + li/2 - 25/ 2 +

H27/ 2 - (22/ 3 + If/2))~4.025

45. ('17 - 4)/8y2 - 1/2

47. f Fxr dx - 10,12 Fxr dx =

- ~,12 Fxr dx is - 1/8 the area of a circ\e of

radius y2 corrected by the area of a triangle (draw a graph).

49. (-2'17cos2'170)/0+(sin2'170)/02. (Tbis tends to zero as 0 tends to 00. Neighboring oscillations tend to cancel one another.)

51. (b) (5e3"./10 - 3)/34 53. (a) Use integration by parts, writing cosnx =

cosn-Ix X cosx. 55. 2'172

57. (a) Q = f EC(a2/w + w)e-O/sin(wt)dt (b) Q(t)= EC{I- eO/[cos(wt) + asin(wt)/wn

59. y

y = [(x) , x = g(y )

[(a ) -f--"-''''--

o b

61. (a) 00 = 2, an others are zero. (b) 00 = 2'17, bn = - 2/ n if n ~ 0, an others are

zero. (c) 00 = 8'172/3, on = 4/n2 if n ~ 0,

bo = 0, bn = - 4'17 / n if n ~ 0. (d) a4 = b2 = b3 = 1, an others are zero.

Review Exerclses for Chapter 7 1. x 2/2 - cosx + C 3. x 4/4 + sinx + C 5. e X - x 3/3 - Inlxl + sin x + C 7. eH + 03/3 + C 9. -cos(x3 )/3 + C

11. e(x3) /3 + C 13. (x + 2)6/6 + C 15. e4x'/12 + C 17. -!cos32x + C

19. x2tan- lx/2 - x/2 + tan- lx/2 + C 21. sin- l(t/2)+t3/3+C 23. xe4x /4 - e4x /16 + C 25. x 2sinx + 2xcosx - 2sinx + C 27. (e-Xsinx - e-Xcosx)/2 + C 29. x 31n3x/3 - x 3/9 + C 31. (2/5)(x - 2)(x + 3)3/2 + C 33. x sin3x/3 + cos3x/9 + C 35. 3x sin2x/2 + 3 cos2x/4 + C 37. x 2e x'/2 - e X ' /2 + C 39. x 2(lnxf/2 - x 2(lnx)/2 + x 2/4 + C 41. 2e'Ix(/i - 1) + C 43. sin x Inlsin xl - sin x + C 45. x tan -Ix ~ In(1 + x 2)/2 + C 47. -I 49. '17/25 51. sin(1) - sin(I/2) 53. ('172/32 + 1/2)tan- I('I7/4) - '17/8

55. (4y2 - 2)/3 + (2y2 - 2)a

57. 3J3/5 59. 399/4

y

40

30

20

13

61. 6 .1'

2.4

63. (2 - y2)/2

y

.<

4 .,.

65. In2

y

67. 2/(n + I) 69. 18.225 71. (a) 90008.46 liters

3

(b) 3000.28 litersiminute

4

73. Hsin('TTxI2)sin('TTx)l'TT + cos('TTxI2)cos('TTx)/2'TT] +C

Chapter 8 Answers

8.1 Oscillations 1. cos(3t) = cos[ 3( t + 23'TT ) ]

3. cos(6t) + sin(3t)

= cos[ 6(t + 23'TT)] + Sin[ 3(t + 23'TT)]

5. cos3t - 2sin3tl3

7. - iß sin(2ß t) 9. 2'TT 13, 3, 1/3

I- :,,/3-l

11. 21T,4, - 1 x

- 4 ~2,,-----l

Chapter 8 Answers A.4S

75. sin - Ix-~ +C 77. (a)(\nx)2/2+C

(b) (2/9)( - ß 13 + I) 79. (xn+llnx n+1 - xn+I)/(n + 1)2 + C 81. (a) (IOO/26)(sin5t/5 + cos5t + e- 25/)

(b) Substitute t = l.OI in part (a). 83. (a) m 2 + n2 + mn + 2m + 2n + 1 = O. (b) The dis­

criminant is negative. (c) Yes; for example X- I / 2

and x(-3 ± .fS)/4.

85. xeOX[b sin(bx) + a cos(bx)]/(a2 + b2) + eOx [(b2 - a2)cos(bx) - 2ab sin(bx)]/(a2 + b2)2 + C

13. -cos2t

x

- I

15. lf6 cos(5t - tan -1(1/5))

x 1.-__ 2# ___ ..,j

- I (I) . 9 tan "5 Phase shlft ; - ; ---

W 5

Period ;~; ~ w 5

17. cos2t + (3/2)sin2t

A.46 ehapler 8 Answers

19. 2cos4x

)'

-2

21. (a) 16'112

(b)

x

x

1 1------1"-------1

-I

0< = Amplitude = 1

23. The frequency decreases by a factor of If. 25. (a) 27(d2x / dt2) = - 3x + 2x3

(b) 27(d2x/dt2) = -3x (c) 6'11

X2 + x,Vkdk l 27. (a) Xo = ----==:-

1+ ~k2/kl (b) !'(xo) > 0

29. There is no restriction on b. 31. Multiply (9)by ",sin",t and (10) by cos",t and

add. 33. (a) V"(xo) > 0, so the second derivative test ap­

plies. (b) ComputedE/dt using the sum and chain

rules. (c) Since E is constant, if it is initially smalI, the

sum of 1. m( dx)2 and V(x) must remain 2 dt

smalI, so both dx / dt and x - Xo remain smalI.

8.2 Growth and Deeay 1. dT/dt - -O.ll(T - 20) 3. dQ/dt - -(0.00028)Q 5. 2e- 3t 7. e3t

9. 2eSt - S 11. 2e6 -2.s

13. 7.86 minutes 15. 2,476 years

y

19. eSt + 1

-2

21. Increasing 25. 33,000 years 29. 1.5 X 109 years 33. 49 minutes 37. 18.5 years

[(t)

4

e 2

2

23. Decreasing 27. 173,000 years 31. 2,880 years 35. 4.3 minutes

39. The annual percentage rate is 1000e rilOO - I) ~ 18.53%.

41. (a) 300 e-O.3t

(b) 2000; 2000 books will eventually be sold. (c)

S(I)

2000 ---- -----

1000

o 10 20 1

43. K is the distance the water must rise to fill the tank.

45. (a) Verify by differentiation. (b) a(t) = t(e- I / t + 1 - e- I )

47. (2m/8)ln2

8.3 The Hyperbolle Funetlons 1. Divide (3) by cosh~. 3. Proceed as in Example 2.

5. fx (coshx) = i fx (eX + e-X) = i (eX - e- X)

= sinhx. 7. Use the reciprocal rule and Exercise 5.

9. (3x 2 + 2x)cosh(x3 + x 2 + 2) 11. cosh x sinh 5x + 5 cosh 5x sinh x 13. - 8 sin 8x cosh(cos 8x) 15. 4 sinh x cosh x 17. - 3 csch23x 19. (2 sech22x)exp(tanh 2x) 21. [sinhx(l + tanhx) - sechx]/(I + tanhx)2 23. (sinhxXj[dx/(l + tanh2x)]) +

coshx/(1 + tanh2x) 25. (sinh 3t)/3

27. 2coshß 1

29. cosh 3t + (sinh 3t)/3 31. 2 cosh 61 33.

y

-3 2 x

35. y

------- 1 -------

37. (sinh 3x)/3 + C 39. Inlsinh xl + C 41. (sinh 2x)/4 - x/2 + C 43. e2x /4 - x/2 + C 45. cosh3x/3 + C

2 x

47. [y - cosh(x + y)]/[cosh(x + y) - x) 49. - 3y sech23xy /(cosh y + 3x sech23xy) 51. (a) Xo = 1

(b) d 2x/dl2 = 2(x - I) 53. Use the definitions of sinhx and coshx. (Don't

expand the nth power!)

8.4 The Inverse Hyperbolle Functions

1. 2x;";x4 +4x2 + 3

3. (3 - sinx) / Y'(3-x-+-co-s-x-)2=-+-1

5. tanh- 1(x2 - I) + 2/(2 - x2 )

7. [(I + I/y?"=l)(sinh-Ix + x)-

(x + cosh-Ix)(1 + I/v?+l)]/(sinh-Ix + xi

9. [exp(l + sinh -IX »)/ v?+l

Chapler 8 Answers A.47

11. -3sin3x;";cos23x + I

13. 0.55 15. 1.87 17. Lety = COSh-IX, so X = HeY + e-Y ). Multiply by

2eY, solve the resulting quadratic equation for eY

and take logs. 19. Let y = sech -Ix so x = 2/(eY + e-Y). Invert and

proceed as in Exercise 17.

21. ~tanh-Ix= d I =--z:: = I Y dx _ tanh y sech'y I - tanh

dy

I

~ sech y

-I

- sech y tanh y

-I

x,.jI - sechy x~ 25. Differentiate the right hand side. 27. Differentiate the right hand side. 29. (l/4)lnl(l + 2x)/(1 - 2x)1 + C

31. (I/2)ln(2x +R+!)+ C 33. In(sin x + ~sin2x + I ) + C 35. (1/2)lnl(l + eX )/(1 - eX)1 + C 37. No

8.5 Separable DiHerential Equatlons

1. Y = sinx + I 3. y = exp(x2 - 2x + I) - I 5. Y = -2x 7. eY(y - I) = (I/2)ln(x2 + I) 9. y = 2x + I

11. y=exp(-sinx)+ I 13.

2.1

1.25

3 4

15. (a) Q = EC(I - exp( - t / RC» (b) 1 = RC In(lOO)

6

17. Verify that the equations hold with dx/dt = 0 and dy/dt = O.

19. P = PoA exp(Pokl)/[1 + A exp(Pokt»)

21. As To increases, COSh( mfox ) ~ I, so Y ~ h, which

represents a straight cable.

A.48 ehapter 8 Answers

23. (a)y'= -y/x (b)y' = X/y;y2 = x2 + C.

25. (a)

(b)y' = 3cx2

(e) y' = -1/3cx2 ; Y = 1/3cx + C 27. (a)

\ \ , y

, \ \ I , \ \ \ , " ' ..... ' ~\ \ ..... ,"'~~\

(b)y = kx2

29. y(l) ~ 2.2469 31. y(l) ~ 0.4683

33. }i,IIJ.,y(x) = 3

y

, I / I I I / I 11 / I 1/ / I" / ,-'­~ ..... '-

4 - - - -

3 - - - - - -/ / / / / /

2 I I I I I I

I I I I I I

I I I I I I , , I , , , 2 3 x

x

35. !im y(x) = I x-->oo

y

6

5

4

3

-I

-2

37. 61

I I I I I I I I I I I I I I

~}J~J!t~}

"""" :/""///////

\\\\\\\\ \ \ \ \

I I I I

39. J h(y)dy = - J (1/ g(x»dx

x

8.6 Linear First-Order Equatlons 1. y = 2 + ( - 31nll - xl + C)(I - x) 3. Y = I + Cexp(x4/4) 5. y = -2 + 2 exp(sin x) 7. y=(eX-e)/x

9. The equation is L ~ + RI = Eoeos wl + EI and

has solution

1= Eo I (!!. sinwl _ weoSWI) L (R/L)2+ W 2 L

E + Ce- tR/ L + ~ R

11. I = EoC - EoC exp( - 1/ RC); 1-+ EoCas1-+ +00.

13. Set y = .9 X 2.51 X Hr and verify the value of I. 15. 6.28 X 105 -(8.28 X 105)exp(-2.67X 10- 11)

- (2.01 X 1 OS)exp( - 1.07 X 10 - 6/ )

17. 15 seconds; 951 meters. 19. Use separation of variables to get

v =/mgjy tanh(hg/m I)

FMo g 2 2 21. - - --(Mo + MI)

Mf 2Mf 23. Y = -2(x + I) + Ce x

y

I \ - /4 I I , /

I \ I

I \ \ \ I

-4 I

4 x , I \ / I , I I \ -/ I I , \ \ ,

/

I , -4 \- I , 25. If YI and Y2 are solutions, prove, using methods of

Section 8.2, uniqueness for y' = P(x)y and apply it to y = YI - Yl. (This is one of several possible procedures.)

27. (a) w' = (l - n)[ Q + Pw]

(b) Y = ± 1/(xFJC2)

29. (a) v = ~ - g(Mo - rt) + C(Mo - rl)y/r-I y - r y - 2r

where C = MJ-y/r( gMo - ~) and y - 2r y - r

where the air resistance force is yv.

(b) At burnout, v = ~ _ gM I + cMy/r-l. y - r y - 2r I

Review Exerclses for Chapter 8 1. y = e3t 5. y = (4e3t - 1)/3

3. y=(I/ß)sinßt 7. y=4/(4-t4)

9. j(x) = e4x

11. j(/) = cosh2t + sinh2t/2 13. X(/) = cos t - sin t 15. x(t) = (sinh 3t)/3 17. Y = -In(l/e + 1- e X )

19. x(t) = e-4t 21. y = - t

23. g(/) = cos(fi73 t - (2/fi73)sin(fi73 I); ampli­

tude is ~19/7 ; phase is - /3fi tan -1(2b/7 )

25.

27. y

7

5

3

-1.648 1----271/w = 4.1--~

I

y

5 x

x

ehapter 8 Answers A.49

y

------- 3 -----

-4 4 x

31. x = e t

33. y = x 2/2 - x - 2e- x + 2 35. y(x)=sinh5x/sinh5 37. 6x cosh(3x2)

39. 2x / ~,....( x-2-+-I-/""'--1

41. cosh 3x / {T+T + 3 sinh 3x sinh -IX

43. (-3/~)exp(l- cosh-I(3x».

45. tan-I(sinhx) + C 47. (l/3)tanh- l (x/3) + C if lxi< 3

(I/3)coth- l (x/3) + C if lxi> 3 49. xcoshx - sinhx + C

51. x(t) = cosJ2.1/5 t 53. (a) k = 640

(b) - 6400 newtons 55. (a) y" + (w 2 - ß)y = 0

(c) x(t) = e-t(coS(ß t) + (l/ß sinß t»

, x

/ /

I '':> I .:-:; , ' , l'

, h r "I ,

57. 66.4 years 59. 54,150 years 61. 27 minutes 63. (a) 73 years

571

3v3

(b) S(t) = ke- at where k = S(O)

A.50 ehapter 9 Answers

65.

EI R -------------

67. (a) y2/9 + x 2 = k, k = 2C /9 y

3 ,. I" 1,,

I , ,

I 'I I I I I , ' , , , 111

-I I I I , , I

\1\ \ , , \ \ \ \ \ \ ,\ ' " .. -3

.. " \ \ \ \ \ \ \ \ \

'" , , , , , I ~ I ,

11 11 , 1 , , , I I , I I , ,

/ " , " ,'/ ,.

x

Chapter 9 Answers f!.1 Volumes by the Slice Method

1. 3'/T 3. Ah/3 5. 2125/54

7. 4,ß /3 9. XI = (1- 3ff74)h, X2 = (1- Vfli)h,

X3 = (1- Vf/4)h 11. 0.022 m3

13. 1487.5 em3

69. 15.2 minutes, no. [Tbe "no" could be "yes" if you allow a faster addition of fresh water after drain­ing.]

71. 1= 2(3 sin '/TI - '/T cos'/T/)/(9 + '/T2) + [1 + 2'/T /(9 + '/T 2)]e- 3t

73. y = -4/3 + Ce3x

" \ \ \

\ \

75. 1

\

y

14

13 1 ~

11

\

\

-1

·- 2

- 3

- 4

f--/ / / x

""" \ \ \ \ \ \

77. Y = e X is the exact solution; y(1) = e ~ 2.71828. 79. Y = -I/(x - 1) is the exact solution, it is not

defined at x = 1. 81. Y = Ce Qt - (b/a); the answers are all the same;. 83. (a) Verify using the chain rule

(b) Integrate the relation in (a) (e) Solve for T = I; the period is twiee the time to

go from (J = 0 to (J = (Jo .

85. (a) y = eosh(x + a) or y = 1. (b) Area under curve equals are length.

15. 38'/T }'

3

x

x

19. 7171/105

) '

23. 1371

\ '

8

6 o 4

4 6 .,.

25. 1371 (See Exercise 11, Section 9,2 for tbe figure,) 27. 5 cm3

29. V = 712(R + r)(R - r)2/4 31. For tbe two solids, A I(X) = A2(x). Now use tbe

slice metbod',

9.2 Volumes by Shell Method

" Je

Chapter 9 Answers A.51

5. 71(17 + 4v'2 - 6/3)/3 y

6

4

2 Je

9. 971 (See tbe Figure for Exercise 23, in tbe left-hand column.)

11. 971 ) ' .

8

6

4

<>

4 6 x

13.471/5 (You get a cylinder when this volume is added to that of Example 5, Sec ti on 9,1.)

15. /3 ",/2 17. 24",2

y

19. (a) V = 4",r2h + "'h 3/3

Je

(b) 4",,2, it is the surface area of a sphere, 21. (a) 271 2a2b

(b) 2",2b(2ah + h2) (c) 471 2ab

23. ",3/4 - ",2 + 2",

A.S2 ehapler 9 Answers

9.3 Average Values and the Mean Value Theorem tor Integrals

1. 1/4

5. 2 9. '/T/2 - 1

13. 9 +ß 17. 55° F 19. (a) x2/3 + 3x/2 + 2

3.ln..ß72

7. '/T/4 11. ~2/3'/T

15. 1/2

(b) Tbe function approaches 2, which is the value of f(x) at x = O.

21. Use the fundamental theorem of calculus and the definition of average value.

23. Tbe average of [J(x) + k) is k + [the averge of f(x»).

\'

-[(x) -+k

"" (x)

'" a

[(x) + k

[(x)

b x

25. f(b) - f(a) = i b j'(x) dx = j'(c)' (b - a), for

some c such that a < c < b.

27. exp [ .eIn fex) dx/(b - a) ]

29. Write F(x) - F(xo) = ff(s)ds. If If(s)1 " M on Xo

[a,b) (extreme value theorem), IF(x) - F(xo)1 " Mix - xol, so given E > 0, let 6 = E/ M.

9.4 Center ot Mass

m.x. + m2x2 + m3x3 m. + m2+ m3

3. Let M. = m. + m2 + m3 and M2 = m4' 5. x=3 7. x = 67 9. x = I, Y = 4/3

11. x = 29/23, Y = 21/23

13. (a) x = 1/2,y =ß /6 (b) x = 3/8,y =ß /8

[ m2x2 + m3x3 + m4x4 ]

15. m.x. + (m2 + m3 + m4) m2 + m3+ m4

m. + (m2 + m3 + m4) m.x. + m2x2 + m3x3 + m4x4

m. + m2 + m3 + m4

17. x = 3 (ln 3)/2, y = 26/27 19. x = 4/(3'/T), Y = 4/(3'/T) 21. x = 4/3,y = 2/3 mx + mx + m x

. •• 2 2 3 3 23. Smce Xj " b, x = + + m. m2 m3

m.b + m2b + m3b " = b. m. + m2 + m3 Similarly a " x. Tbe center of mass does not lie outside the group of masses.

25. Differentiate x to get the velocity of the center of mass and use the definitions of P and M.

27. x = -4/21, Y = 0

y

x

29. x = (Jin/4 - 1)/(Ji - 1), Y = 1/[4(Ji - 1») 31. x = (XI + x2 + xN3,y = (YI + Y2 + Y3)/3

Supplement to 9.5: Integrating Sunshine

1. The arctic circle receives 1.25 times as much en­ergy as the equator.

364 3. (a) F = ~ ,{ Jcos21- sin2D +

T-O

sin I sin D cos-·( - tan ltan D)}

(b) Expressing sinD in terms of T, the sum in (a) yie1ds

.J;'" {J'''~ -"n'. ,,,'(2.T /365)

+ sin I sin a cos(2'/TT /365)

X cos - I [ - tan I sin a cos(2'/TT /365) ]}dT.

)1 - sin2a cos2(2'/TT /365)

This is an "elliptic integral" which you cannot evaluate.

5. nsinlsinD 7. 0.294; it is consistent with the graph (T = 16.5;

about July 7).

9.5 Energy, Power, and Work 1. 1,890,000 joules 3. 360 + 96/7T watt-hours 5. 3/2 7. 0.232 9. 98 watts 11. (a) 18t2 joules

13. 1.5 joules

17. 41,895,000 joules 21. 0.15 joules

(b) 360 watts 15. (a) 45,000 joules

(b) 69.3 meters/seeond 19. 125,685,000 joules 23. 1.48 X 108 joules

Review Exereises for Chapter 9 1. (a) n2/2

(b) 2n2

5. 64.j2 7T /81 9. 5/4

13. 6

3. (a) 3n/2 (b) 2n(21n2 - 1)

7. 72n/5 11. I

15. Apply the mean value theorem for integrals.

17. 1/3,4/45,215 /15

19. I, (e 2 - 5)/4, ~ /2 21. 3/2, 1/4, 1/2

23. (a) 7Tfp(X)[f(x)fdx

(b) (l47T /45) grams 25. x = 5/3,y = 40/9 27. x = I / 4(2 In 2 - I), Y = 2(ln 2 - IN (2 In 2 - I) 29. x = 27/35, Y = -12/245 31. (a) 7500 - 21OOe- 6 joules

(bH(l25 - 35e- 6 ) watts

Chapter 10 Answers 10.1 Trigonometrie Integrals

1. (eos6x)/6 - (eos4x)/4 + C 3. h/4 5. (sin 2x)/4 - x/2 + C 7.1/4- 7T/16 9. (sin 2x)/4 - (sin 6x)/ 12 + C

11. 0 13. -1/(3eos3x)+ 1/(5eos5x)+ C 15. The answers are both tan -IX + C

17. Jx2 - 4 - 2eos-1(2/x) + C

19. (l /2)(sin -lU + u[f"7) + C

21. ,f4+? +C

23. (-1/3)F? (x2 + 8) + C

25. (1/2)sinh- I«8x + 1)/[i5) + C

27.

__ 1_ Inl6X + I + 6,[3 [i3

(6x + 1)2 I 13 -I +C

Chapter 10 Answers A.53

33. 120/ 7T joules

35. pgw fx2[f(a) - f(x)]j'(x)dx; the region is that

under the graph y = fex), 0.;;; x .;;; a, revolved about the y-axis.

37. (a) The force on a slab of height fex) and width

dx is dx fo!(X)pgy dy = t pg[f(x)f dx. Now in­

tegrate. (b) If the graph of fis revolved about the x axis,

the total force is pg/27T times the volume of the solid. 2 (e) "3 Pg X 106 = 6.53 X 109 Newtons.

(e) Show that if the standard deviation is 0, k; - /L = 0, whieh implies k; = /L.

(d) {l ± [ai -l ± a;]2} 1/2 n j=1 n ;=1

(e) All numbers in the list are equal. 41. Let g(x) = f(ax) - c. Adjust a so g has zero inte­

gral. Apply the mean value theorem for integrals to g. (There may be other solutions as weil.)

43. The average value of the logarithmie derivative is ln[f(b)/f(a)l/(b - a).

29. 1,0,1/2,0,3/8,0,5/16.

31. x=(ß -.j2)/ln«ß +2)/(.j2 + 1»-1

y = (tan- 12 - 7T/4)/[2In«ß + 2)/(.j2 + I))] 33. 125

35. ,[3,9.j2 /4 37. (a) Differentiate [S(t)]3 and integrate the new ex­

pression. (b) [3(-teost + sint + t/8 - (l/32)sin4t)]1/3 (e) Zeros at t = nw, n a positive integer. Maxima

oeeur when n is odd.

10.2 Partial Fraetions 1. (l / l25){ 41n[(x 2 + 1)/(x2 - 4x + 4)] +

(37 /2)tan -Ix + (15x - 20)/2(1 + x 2)_

5/(x-2)}+C 3. 5/4 - 37T/8 5. (I/5){ln(x - 2i + (3/2)ln(x 2 + 2x + 2)­

tan-I(x + I)} + C

A.54 ehapler 10 Answers

7. 2 + (l/3)ln3 + (2/ß)(tan- I(5/ß)­

tan -1 (3J3» 9. (I/S)ln«x2 - 1)/(x2 + 3» + C

11. (I /2)ln(5/2)

13. 2..[X - 2 tan -I..[X + C 15. i(x2 + 1)4/3 + C 17. -2/(1 + tan(x/2» + C 19. w /16 - (1/4)lnl(l + tan(w /S»(I + 2 tan(w /S)­

tan2(w/S»I~ -0.017 21. w In(225/ 176)

23. 3(1 + Xf/3/4 + (3/4 V4 )lnl 3l4 (I + x)2/3 + (2 + 2X)I/3 + II - (1/2 \1432)tan -1[(2(4 + 4X)I/3

+ Vi /6"fiOS] + C

2 Illx-SOI 1 1 4 5. (a) 20 n x _ 60 = kt + 20 n"3

SO(I - e- 20kl )

(b) x = 4 -20kl ]-e

(c) 26.2 kg

27. (a) Using the substitution, we get

(q/m) J up+q-1x,-m+ldu.

(b) If r - m + I = mk, the integral in (a) becomes

(q/m) J uP+q-l(u q - b)k du

which is an integral of a rational function of u.

10.3 Are Length and Surfaee Area

1. 92/9 3. 14/3

5. Lb b + n2x 2n 2 dx

7. .r b + cos2x - 2x sin x cos x + x 2sin2x dx

9. ,(5 +[I +yTO

11. ,(5 +[I +.ff7

13. (w /6)(133/ 2 - 53/ 2) 15. 2654w /9

17. 2w([I + In(1 + [I» 19. w[(34/ 3 + 1/9)3/2 - (10/9)3/2]

21. 2[I w

23. w(6[I + 4,(5) 25. (l/27a2)[(4 + 9a2(l + b»3/2 - (4 + 9a2W/2]; the

answer is independent of c.

27. J2 b + 36x4 dx~ 19 -)

29. (a) fo'IT/2b + sec4x + 4sec2x dx

(h) 2w fo'IT/2(tanx + 2x)Jr-5-+-se-c"'-4x-+-:-4-se~c2r-x dx

31. (a) f JI + (I - 1/ X 2)2 dx

(b) 2w f(l/x + x)~ri -+-(I-_-I/-x-2)~2 dx

33. Dividing the curve into I mm segments and re-volving these, we get about 16 cm2.

35. Use Isinß xl < I to get L <f'IT v'I'TI dx = 4w.

37. Estimate each integral numerically.

39. 2w Lb[I/(l + x 2)]( ~I + 4x2/(1 + X 2)4 )dx; the

integrand is <,(5/(1 + x 2).

41. (a) w(a + b)~ (b - a) (b) Use part (a).

10.4 Parametrie Curves 1. y = (I/4)(x + 9)

y

3. I = (x - 1)2 + l y

x

x

5. x = t,y = ±JI=2t2 or x = cost/j2,y = sint 7. x = t,y = 1/41. 9. x = I, Y = 13 + 1.

11. x = t, Y = cos(2t). 13. y = (1/3)(x + 3/2) 15. y=I/2 17. (13, -7)

y

- I

19. Y = eos/X (x > 0), horizontal tangents at 1 = n7T,

n a nonzero integer. The slope is - 1/2 at 1 = 0 although the eurve ends.

21. y2 = (l - x)/2, vertieal tangents at 1 = n7T, n an integer

y

x

23. (133/2 - 8)/27

25. (1/2)(.;5 + (l/2)ln(2 +.;5)] 27. (a) Calculate the speed direetly to show it equals

lai-(h) Calculate direetly to get lal(/1 - 10)

29. (a) y = -x/2 + 'TT/2 - I

2 x

(e) f' {5 - 3 eos2(J - 2eos (J d(J

31. 5 33. (a) x = k(eos wl - wl sin w/);

j = k(sin wl + wl eos w/). (h) k{1 + w2/ 2

(e) 2mkw 35. (a) x = 1 + (l + 4/2)-1/2,

Y = 12 + 2/(1 + 4/2)-1 / 2

(b) x = ±(1/2)...j1/(x2 - y) - 1 + ";x2 - y.

x

Chapler 10 Answers A.55

37. (a) We estimated about 338 miles. (h) We estimated about 688 mi1es. (e) It would probably he longer. (d) The measurement would depend on the defini­

tion and seale of the map used. (e) From the World Almanac and Book 0/ Facls

(1974), Newspaper Enterprise Assoe., New York, 1973, p. 744, we have eoastline: 228 miles, shoreline: 3,478 miles.

10.5 Length and Area in Polar Coordinates

1. 12/2

3. (4/3)(133/ 2 - 8)

5.9'TT/4

Ji 3

y

1f x

9. 33'TT/2

Ji

- 4 4 x

3

11. 2'TTr

A.56 Chapter 10 Answers

13. S =J"'/2 ~sec\0/2)/4 + tan2(0/2) dO -",/2

A =2 - ",/2

y

r

15. s = 10.,,/4 ~sec2fj tan2fj + sec2fj + 4sec 0 + 4 dO

A = 1/2 + ",/2 + In(3 + 2/2)

y

17. s =

/

/ /

/

;

r <-i-> = 3.414

2 3 x

1o'1t/2~(l + cosO - 0 sin 0)2 + 02(1 + 2 cosO + cos~.) dO

A = (1/2)[",3/16 + ",2/2 - 4 - ",/8]

19. s = 1o"'/2J(5 + 4 sin 20)/(1 + 2 sin 20) dO

A = ",/2 21. A = (1/4X5",/6-ß)

L = (2 +ß)",/6

y

23. A = ",/2 L = 2", + 8

y

x

25. /2(e2(n+I)'1T _ e211'lT)

27. (a) Use x = acost,y = bsint, where T= 2",. (b) Substitute into the given formula.

Review Exerclses for Chapter 10 1. sin3x + C 3. (cos2x)/4 - (cos8x)/16 + C 5. (1- X 2)3/2 -F? + C 7.4(x/4-tan- l(x/4»+C 9. (2tr /7)tan- I[(2x + I)/tr] + C

11. Inl(x + I)/xl-I/x + C 13. (1/2)[lnlx2 + I1 + 1/(x2 + I)] + C 15. tan-I(x + 2) + C

17. - 2jX cosjX + 2 sinjX + C 19. -(1/2a)cot(ax/2) - (1/6a)cot3(ax/2) + C 21. Inlsecx + tanxl- sinx + C 23. (tan-lxi /2 + C .

25. (l/3W)[lnlx- WI-ln~x2+ Wx+3 3ß +ßtan- I«2x/W + I)/ß)]+ C.

27. 2jX e./x - e./x + C 29. x - In(e X + I) + C 31. (-1/4)[(2x2 - 1)/(x2 - If] + C 33. -(1/1O)cos5x - (l/2)cosx + C

35. In~ +C 37. 2e'1x + C 39. tln2 41. tln(x2 + 3) + C 43. x4lnx/4 - x 4 /16 + C 45. H(ln6 + 5)4 - (ln 3 + 5)4]R:: 186.12 47. (1/4)sinh2 - 1/2 49.0 51. (7333/ 2 - 43/ 2)/243 53. 59/24 55. ",(53/ 2 - 1)/6 57. R:: 31103

59. x = (y + 1)2

y

Chapter 10 Answers A.57

89. (a) b - a + (b n+1 - an+I)/(n + I) if n -:!= -I. If n = -I, we have b - a + In(b/a).

(b) n = 0: L = b - a; n = 1: L = {2 (b - a);

n = 2: see Example 3 of Section 10.3; for n = (2k + 3)j(2k + 2), k = 0, 1,2,3, ...

-1 { I/(I-n) k (_I)k- j }IX=b

L = _n. __ (I + n2x 2n - 2)3/2 2: (~). (I + n2x 2n - 2'/ n - I j=o] 2] + 3

x=a -2

61. Y = 2x /3 + 1

63. x = O,y > 0

y

x

n = t : L = ir[(4 + 9b)3/2 - (4 + 9a)3/2].

(e) Around the x-axis, we have

[ 2(bn+ 1 - an+l) b2n+ 1 _ a2n+ 1 ]

w b - a + n + 1 + 2n + 1

if n -:!= - I or - 1/2. For n = - 1 we have

w[ b - a + 2In(b/a) - (a- I - b-I)J.

For n = -1/2 we have

'17 [ b - a + 4yb - 4{ti +In(b/a)]­

Around the y-axis we have

[ 2(bn+2 _ a n+2) ]

w b2 - a2 + ----,--;;--­n+2

if n -:!= -2. For n = -2, we have 'I7[b 2 - a2 + 2ln(b / a)].

(d) A", = 2wL (from 89(b» + Ax (from 88(d» Ay = Ay (from 88(d»

Some answers from 88(d) needed here are: 88(d).

n = 0; Ax = 2'17(b - a)

n = I; Ax ={2 w(b2 - a2)

x=b n = 2; Ax = ;; [(I + 8x2)2xv'1+'47 -ln(2x +v'1+'47) JL=a

65. Y = 3x/4 + 5/4

67. (1/8)(,;B7 . 16 + Inl,;B7 + 161)

w 4 3/2I x - b n = 3; A x = 27 (1 + 9x) x=a

n = (2k + 3)j(2k + 1); k = 0, 1,2,3, ... 69. L=(1/3)[(w2/4+4)3/2_8] A = wS /320

71. L = rJ'(5-/-4-) -+-e-os-2-0-+-3-si-n2:-2-0 dO k (l)k- j

A x = ~n(l+n)/(I-n)(1 + n2x 2n - 2)3/2 L (~) -. (1 + n2x 2n - 2'1 n - 1 j=O] 2] + 3

A = 3'17/8 73. L = 5{2

A = 315w/256 + 9/4 75. b2 = I, an others are zero. 77. a3 = I, all others are zero. 79. a4 = 3, an others are zero. 81. ao = I, a2 = -1/2, an others are zero. 83. (a) (1/ k2)In[NoCkIN(t) - k2)/ N(t)(kINo - k2)]

(b) N(t) = k, No/[k INo(1 - ek2') + k2ek2']

(e) The limit exists if k2 > 0 and it equals k2/k l •

85. Use (eOScf»dcf> = (sincf>m)(eos ß)dß and substitute.

87. a- I/2InI2ax + b + 2 {ti ~ax2 + bx + c 1 + C, a>O

(- a)-1/2sin- I[( -2ax - b)!Vb2 - 4ac] + C, a<O

n = 0; Ay = w(b2 - a2)

n = I; Ay ={2 w(b2 - a 2)

n = 2; Ay = ~ [(1 + 4b2)3/2 - (I + 4a2)3/2]

n = (k + 2)/(k + I); k = 0, 1,2,3, ... ;

A = ~n2/(I-n)(1 + n2x 2n - 2)3/2 Y n - I

k (l)k- j

X 2: (~) -. (I + n2x2n-2'11~ j=o] 2] + 3

91. (a) 2w LßrsinO~r2 + (r,)2 dO

(b) 2wJ"/4 eos20 sin ob + 3sin220 dO -,,/4

A.58 ehapler 10 Answers

93. (a)

m= I,n= I

y

I

m=I,n=2

y

(b) Each curve will consist of n loops for n odd or even.

(c)

-I

(d)

-I

m = 2,n = I Y

m= 2.11 = 4

Y

m = 3,n =4

y

x

m = 2,n = 2

y

I

-I

m = 2, n = 5

Y

m = 3,n = 5

r

-I

95. The last formula is the average of the first two.

x

m= 1,11=3

y

m = 2, n = 3

y

m=I,n=4

y

I x

Chapter 11 Answers 11.1 limits of Functlons

1. Choose () less than land e/(I + 21al). 3. Write x 3 + 2x2 - 45 =

[(x - 3) + W + 2[x - 3f - 45 and expand. 5. e3 7. 5 9. -4 11. 6

13. A = I/Vi 15. A =-lne/3 17. - 2 19. 2/3 21. 3/5 23. 1/2

25. O. Consider ~ - x as the difference be­tween the hypotenuse and a leg of a right triangle. As x gets large, the difference becomes small.

27. y = - I is a horizontal asymptote.

y

-4 2 4

-I

29. + ao 31. + ao 33. + 00 ~5. - 00

37. -I 39. -I 41. Vertical asymptotes at x = 2, 3. Horizontal

asymptote at y = o.

x

43. Vertical asymptotes at x = ± I, horizontal asymptote at y = O.

)1 y

x

x

45. (a) Given e, the A for g is the same as for f (as long as /g(x)/< /f(x)/ for x;;. A).

(b) 0 47. 7/9 49. 3/2

~ I

Chapler 11 Answers A.59

51: 4/5 55. 16/17 57. + 00

53. 2n + I

59. - 00

61. Y = 0 is a horizontal asymptote; x = - I, x = I are vertical asymptotes.

63. y = ± I are horizontal asymptotes. 65. If fex) = anx n + . .. and g(x) = bnx n + ... ,

show that an / bn = I. If 1 = ± 00, then limx ->_ oof(x) can be ± limx->oof(x).

67. (a) rex) = - I for x < O,j'(x) = I for x > O,j'(O) is not defined.

!'(x)

x -----Q-l

(b) As x~O -, the limit is -I, while as x~O +, we get 1.

(c) No. 69. (a)

y y

x

(i) a poor 6 (ii) a good 6

y

"~'-'~-",r-'---, .v-B I y=B

Y = [(xl

(j) a poor 6 Iii) a good 6

71. No, which means that the population in the distant future will approach an equilibrium value No.

73. Use the laws of limits 75. Write af(x) + bg(x) - aL - bM =

a[f(x) - L) + b[g(x) - M)

A.60 ehapler 11 Answers

77. Repeat the argument given, using Ix - xol < ö in place of Xo < x < Xo + ö.

79. Given B > 0, let e = 1/ B. Choose ß so that II/j(x)1 < e when Ix - xol < ß; then If(x)1 > B for Ix - xol < ß.

81. If x;;;' A,y < ß where ß = 1/ A,y = I/x.

11.2 L'H6pltal's Rule 1. 108 5. -9/10 9. 00

13. 0 17. I 21. 0 25.0

3.2 7. -4/3

11.0 15.0 19.0 23.0

27. does not exist (or is + 00)

29. 0 31. 1/24 33. 0 35. 1/120 37.0 39. The slope of the chord joining (g(a), f(a» to

(g(b), f(b» equals the slope of the tangent line at some intermediate point.

41.

43. (a) 1/2 (b) I (c) yes

I/I! x

11.3 Improper Integrals 1. 3 5. (ln3)/2 9. Use l/x3

13. Use l/J3x on [I, 00) 17. 3 3{fö 21. Diverges 25. Converges 29. Converges 33. Converges 37. Diverges 41. k > I or k = 0

3. e- s /5 7. 'Ir/2

11. Use e- X

15. Use I/x

19. 2 23. Converges 27. Converges 31. Diverges 35. Converges 39. Diverges 43. ~2.209

45. 6,ß hours 47. 'lre- 20/2 49. ln(2/3) 51. Follow the method of Example 11.

53. (a) Change variables (b) Use the comparison test. (Compare with e X / 2

for x< -I and e- x / 2 for x;;;. 1.) 55. (a) 'Ir

(b)(p - I)(q - I) < O.

57. f(x) = f(O) + fox!'(s)ds; the integral converges.

11.4 Limits 0' Sequences and Newton's Method

1. n must be at least 6. 3. limn-+oo(an) = 2 5. 0, -1,4 - 2v'2, 9 - 2,ß, 12 7. 1/7,1/14,1/21,1/28,1/35,1/42 9. The sequence is 1/2 for all n.

11. N ;;;. 3/e 13. n;;;' 3/2e 15. 3 17. -3 19. 4 21. 0 23. 0 25. The limit is I. 27. The limit is I. 29. 0 31. 0 33. 0 35. (a) x = 0.523148 is a root.

(b) x = -0.2475,7.7243 37. x = 1.118340 is a root. 39. One root is x = 4.493409. 41.

a=2

AI 1.1656

a =3

1.3242 A2 4.6042 ·4.6407 A3 7.7899 7.8113

43. 1/ e R:l 0.36788 45. a. = 220 - 1

a=5

1.4320 4.6696 7.8284

47. Use the definition of limit and let e be a. 49. 1,1/2,1/4,1/8,1/16, ... ,1/(2·), ... ; the

limit is O. 51. The limit does not exist. 53.3/4 55. (a) For any A ;;;. 0 there is an N such that a. ;;;. A

if n ;;;. N, (b) let N = 16A. 57. (a) Assurne lim bn < Land look at

.-->00

lim b. - lim an' n ....... oo n-+oo

(b) Write bn - L = (bn - a.) + (an - L) < (cn - an) + (an - L).

59. (a) Below about a = 3.0, iterates converge to a single point; at a ~ 3.1, they oscillate be­tween two points; as a increases towards 4, the behavior gets more complicated.

(b),(c) See the references on p. 548.

11.5 Numerical Integration 1. 2.68; actual value is 8/3 3. ~0.13488 5. ~ 0.3246 7. ~ 1.464

9. R:; 2.1824 11. Evaluation gives a (x~ - xÖ/3 + b(x~ - xf)/2 +

C(X2 - X2)' Sinee j""(x) = 0, Simpson's rule gives the exaet answer. The error for the trapezoidal rule depends on rex) and is nonzero.

13. 180, 9 15. 158 seeonds 17. The first 2 digits are eorrect.

Review Exerclses for Chapter 11 1. Choose ß to be min(I, e/4). 3. Choose b to be min(I,e/5); min{l,e/3) is also

correct. 5. tan(-I) 7. I 9. 0 11. 00

13. 0 15. 0 17. Y = ± 71 /2 are horizontal asymptotes.

y

" -------------T ----------

21. 1/4 25.0 29. -1/6 33. I 37.0 41. e2

ehapler 12 Answers A.61

23. 0 27. 5 31. see2(3) 35.0 39. I 43.0

45. Converges to I 47. Diverges 49. Converges to 2 51. Converges to 5/3 53. Converges to -1/4 55. 271/3.[3 57. 71/4 59. 32,768 61. e8 63. 0 65. I 67. tan 3 69. Does not exist 71. -2/5 73. I 75. 0 77. -1.35530 (the only real root)

79. 1.14619 81. 2.31992 85. Both 89. (b)

91. I

83. 50.154 87. I/IX

93. Sn is the Riemann sum for fex) = X + x 2•

95. The exaet amount is

X P(e' + e364,/365 + ... + e'/365)

" ----------------- 2:

19. Y = 0 is a horizontal asymptote.

y

-2 -I

2 3 x

Chapter 12 Answers 12.1 The Sum of an

Infinite Serles 1.1/2,5/6,13/12,77/60 3. 2/3,30/27,38/27,130/81 5. 7/6 7.7 9. $40,000 11. 1/12

13. 16/27 15. 81/2 17. 3/2 19. 64/9 21. LI diverges and LI /2; eonverges

97. (a) y

~Y=f(X)

~

(e) Choose ß = e/2m, (or h, whiehever is smallest). 101. (a) Use the definition of N

(b) Use the quotient rule (e) IN(x) - xl < (Mq/p2)lx - xl2

(d) 5

23. 7 25. Diverges 27. Diverges 29. Diverges 31. Reduce to the sum of a eonvergent and a diver­

gent series. 33. Leta;=1 andb;=-1. 35. (a) al + a2 + ... + an = (b2 - bl ) +

(b3 - b2) + ... + (bn + 1 - bn ) = bn + 1 - b l

(see Section 4.1). (b) 1

A.62 Chapter 12 Answers

37 (b) "=' 12/27 and "='( _ r' 12/13 • ",12n+ 1 = T-=-r '" 2n+2 - I - r

The sum is 1.

12.2 The Comparlson Test and Alternating Serles

1. Use 8/3 i

5. Use 1/3i

9. Use I/i 13. Converges 17. Diverges 21. Converges 25. Converges 29. Converges 33. Converges 37. 0.37 41. Diverges 43. Converges absolutely 45. Diverges

3. Use 1/3i

7. Use 1/2i

11. Use 4/3i 15. Converges 19. Converges 23. Diverges 27. Diverges 31. Diverges 35. 0.29 39. Diverges

47. Converges eonditionally 49. Converges eonditionally 51. -0.18 53. -0.087 55. Converges

57. (a) al = 2, a2 = [6, a3 = ~4 + [6

(b) lim an ~ 2.5616 n--+oo

59. Inereasing, bounded above. (Use induetion.) 61. Inereasing for n ~ 2, bounded above. 63. Show by induetion that a2' a3' . .. is deereasing

and bounded below, so eonverges. The limit I

satisfies I = t (l + ~ ). 65. nli.~ an = 4

67. The limit exists by the deereasing sequenee prop­erty.

69. Compare with (3/4t.

12.3 The Integral and Ratio Tests 1. Diverges 5. Converges 9.0.44

13. Converges 17. 11.54

3. Converges 7. Converges

11. Use Figure 12.3.2. 15. Converges 19. (a) ~ \.708

(b) ~ 1.7167 (e) 8 or more terms.

21. Converges 23. Diverges 25. Converges 27. Diverges 29. Converges 31. Converges 33. Converges 35. Converges 37. Show that if lal ln > I, then lanl > 1. 39. p> I 41. p> I

I n-I I 43. (a) S - "2 Jen) = .~ j(i) + "2 Jen) +

1=1

-21 in+lj(x)dx +iOO j(x)dx n n+1

.;;; i~/(i) + t Jen) + t Jen) + i:/(x)dx;

now use the hint. (b) Sum the first 9 terms to get 1.0819. The first method saves the work of adding 6 additional terms.

45. (b)

S

SI(r) --­

S2(r) -

S3(r) ----

SI (0) = I

82 S2(0) = 81= 1.012

51331 S3(0) = 50625 = 1.014

12.4 Power Serles 1. Converges for - I .;; x < 1. 3. Converges for - I .;; x .;;; 1. 5. Converges for 0 < x < 2. 7. Converges for all x. 9. Converges for - 4 < x < 4.

11. R = 00

13. R = 2 15. R = 00

17. R = I, eonverges for x = 1 and - I. 19. R = 3 21. R = 0 23. Note that j(O) = 0 and 1'(0) = I. 25. (a) R = I

(b) Lf..IX i + 1

(e) j(x) = x(2 - x)/(l - xf for lxi< I (d) 3

27. L~=o[( - Itx 2n / n!) 29. tan -I(x) = L~=O[( _1)nx 2n+ 1/(2n + I)], and

(d/dx)(tan -lX) = L~=o( -1)"x2".

31. 1/2 + 3x/4 + 7x2/8 + 15x3/16 + ... 33. x2 - x4 /3 + 2x6 /45 + ... 35. Set fex) = 1/(1 - x) and g(x) = -x2/(1 - x). 37. (a) x + (1/3)x3 + (2/15)xS + ...

(b) I + x2 + (2/3)x4 + ... (e) 1- x2 + (1/3)x 4 - •••

39. Li"=I(-IY+I(l/i)x i

41. Use the fact that i..[i ~ I as i ~ 00.

43. Write j(x) - j(xo) = (j(X) - .± aix i) 1-0

+ ( .~ aix i - .± aix&) + ( ± aix& - j(xo») 1-0 1-0 1-0

45. Show that f(x) = fox g(t)dt.

12.5 Taylor's Formula 1. 3x - 9x3/2 + 8lxs/40 - 243x7/560 + ... 3. 2 - 2x + 3x2/2 - 4x3/3 + 17x4/24 - 4x5/15 +

7x6/80-8x7/315+ ... 5. 1/3 - 2(x - 1)/3 + 5(x - If /9 + O· (x - 1)3 7. e + e(x - I) + e(x - 1)2/2 + e(x - 1)3/6. 9. (a) I - x2 + x6 + ... (b) 720

11. Valid if -I< x" I (Integrate 1/(1 + x) = I - x + x 2 - x 3 + .... )

13. Let x - I = u and use the bionomial series. 15. (a) 1- (1/2)x 2 + (3/8)x4 - (5/16)x6 +

(35/128)x 8 - •••

(b) (-1/2)( -1/2 - I) ... (-1/2 - 10 + I)· (20!)/(1O!)

17. fo(x) = fI(x) = I, f2(X) = hex) = 1- x2/2, Ux) = 1- x 2/2 + x 4/24.

--t---_...+o __ ---.. -y = 1

19. 1:::::10.095 21. 1:::::10.9 23. 1:::::10.401

x

25. (a) Tbe remainder is less tban R 4M3/12 where M3

is tbe maximum value of 1f'''(x)1 on tbe inter­val [xo - R,xo + R).

(b) 0.9580. Simpsons rule gives 0.879. 27. -4/3 29. 1/6 31. ~~_oXn for lxi< I 33. ~~_o2x2n+ I for lxi< I 35. ~~_ox2n for lxi< I 37. 1 + 2x2 + x 4

39. J.xln t dt = ~~-2{( -1)i(X - I)i/[i(; - I))}.

x Inx = (x - I) + ~f .. 2{( -1)i(X - I)i /Ii(i - I))}.

Conclude J.xlntdt = xlnx -x + 1.

41. 1,0,1/2,0 43. 0, -1,0, -1/2 45. 1/2 - x 2/4! + x 4/6! - ... 47. I - 2x + 2x2 - 2x3 - 2x4 + 2xs + ... 49. (a) (x - I) - (x - 1)2/2 + (x - 1)3/3 -

(x -IN4 (b) I + (x - e)/ e - (x - e)2/2e2 +

(x - e)3/3e3 - (x - e)4/4e4

(e) In 2 + (x - 2)/2 - (x - 2)2/8 + (x - 2)3/24 - (x - 2N64

51. In2 + x/2 + x 2/8 - x 4/192 + ...

Chapler 12 Answers A.63

53. sin I + (eos I)x + I(eos I - sin 1)/2)x2-[(sin 1)/2)x3 + ...

55. (a) 0.5869768 (b) It is witbin 1/1000 of sin36°. (e) 36° = '1T /5 radians, and sbe used the first two

terms of tbe Taylor expansion. (d) Usethefaet that 100 = 11:/18 radiansand tanx ~

x(l + rf3) 57. (a) 0, -1/3,0

(b) 1- x 2/3!+x4/5!-x 6/7!+ ... 59. Follow the method of Example 3(d).

12.6 Complex Numbers 1. -; 3. -; 5.

y

2 •

7. y

3

9. y

11. y

0.2 •

0.1

13. -14 + 8i 17. (5 + 3;)/34 21. ±j3i 25. (7 ± ,[53)/2 29. ±2,f2(; -.1)

x

x

4 x

x

15. 3 + 4i 19. (41 + 3;)/65 23. (I ± /ff i)/6 27. ±2(l + i) 31. -I

A.64 ehapler 12 Answers

33. -11/5 35. 328/565 37. 5 - 2i

39.13 -i/2 41. -1/3 + 2i/3 43. (-7 + lIi)/20 45. 3

47·lzl={2,9=5'lT/4 y

51(/4

x

49. Izl = 2,9 = 0

y

51. Izl = 5/6, 9 = -0.93 y

53. Izi ={f4, 9 = 2.19 y

-5

x

55. Izl =.f68, 9 = - 2.9 or 3.4 y

-8

x

57. Izl = ~1.93 , 9 = 0.53 y

0.7

1.2 x

59. Let z. = a + ib, Z2 = c + id and calculate IZ.Z21 and Iz.I·lz21.

61. (8 + 3i)4 63.

y

65.

-3+IOi Y

(-3,4)

67. 2/5

", I \ I \ I \ I I

71. 1/2+ßi/2 75. ei

x

69. ex,y 73~ -ei 77. (3 - 4i)/25

79. (a) e ix • e- ix = (cosx + i sin x)(cos x - isinx);

multiply out (b) Show e Z • e- Z = 1 using (a).

81. Show e3"i/2 = - i. 83. Use (e i9 )n = e in9.

85. {2 eW / 4

87. (/5 /5)e i(O,46)

89 • .ß8 e i ( -0.4)

91. (m /2)e i(-1.74)

93. 25e i( 1.8S)

95. e i(7T/1S+27Tk/S), k = 0, 1,2,3,4; (1.08) e i(O.22+27Tk/S),

k=0,1,2,3,4

y

x

y

fTl. l-C1Eei(O.lS5+2~kI6), k = 0,1,2,3,4,5; 1~i(O.107+2!tk16), k = 0, 1,2,3,4,5

y

x

y

x

99. z is rotated by 7T / 4 and its length multiplied by

1/12. 101. Show that Z4 = 1 and then that z2 = I. 103. Write e i9 = cosfJ + isinfJ.

105. i (I2Z + I -;-/12 - I) ~12 -1

X(l2z- I +i~I2-I) ~12 -I

107. (z + 2i + 2)(z - 2) 109. (a) tanifJ = itanhfJ (b) tanifJ = (tanhfJ)e i7T/ 2 111. Zl = aiz2 , a areal number

Chapler 12 Answers A.65

113. (a) Factor zn - 1 (b) Use your factorization in (a). (c) -I,i,-i

115. The motion of the moon with the sun at the origin. 117. (a) (2n + I)xi for any integer n.

(b) You could define In( -I) = hr, aithough there are other possibilities.

12.7 Second-Order Linear Differential Equatlons

1. y = clexp(3x) + C2exp(X) 3. y = clexp(x/3) + C2exp(X) 5. y = texp(3x) - texp(x) 7. y = e X

9. y = clexp[(2 + i)x) + c2exp[(2 - i)x) = exp(2x)[alcosx + a2sinx)

11. y = clexp[(3 + 2i)x) + c2exp[(3 - 2i)x) = exp(3x)[olcos2x + o2sin2x)

13. y = x exp(3x) 15. y = (x - l)exp(-12 +12 x) 17. (a) Underdamped

(b) x = (l/w)(sinwt)exp(-7Tt/32), w= w-/ill /32 R:: 7T /2.

x

19. (a) Critically damped (b) x = texp( - 7Tt/6)

x

-3 123456

-2

-3

21. Y = clexp(3x) + C2exp(X) + 2x + 6. 23. x = c,exp(t/3) + C2exp(t) + (2/S)cost +

( -I/S)sint 25. y = e2x(c,cosx + C2sinx) +

x 2/5 + 13x/2S + 42/125 27. Y = (c, + c2x)exp(12 x) +

[(I + 212)/9)cosx + [(I -12)/9)sinx

-r

A.86 ehapler 12 Answers

29. Y = clexp(3x) + C2exp(X) + 2x + 6 31. x = c,exp(t/3) + C2exp(t) - sint/5 - 2rost/5 33. y = clexp(3x) + C2exp(X) +

[exP(3x/2)]!(tanx)exp( -3x)dx-

[exp(x)/2]! (tanx)exp( - x)t/x

35. y = e2x(C,cos2x + C2sin2x) + [e2xros2x/2]· J (eh[(1 - cot2x)(cos2x)-

(I + rot 2x)(sin 2x)] . (l + ros2x)} -I dx + [e 2Zsin2x/21J (e2X [(1 - tan2x)' (cos2x) + (I + tan2x)(sin2x)](1 + cos2x)} -It/x

37. x= -cos2t + cost=2sin(3t/2)sin(t/2) 39. x = (-1/24)cos5t + (1/5)sin5t + (l/24)cost 41. (a)

x(t) = e-4t[ ~ sin(.f2f t) - 54025 cos(.f2f t)] 101.f2f

+ ~ cos[ 2t - tan- I ( ~.)] (b) Looks like (2/.[5ö5)ros(2t - tan- I(8/21»

43. (a) x(t) = exp( - t/2)[(7 /IO)cos(!f5 t/2) + (-1/2!f5)· sin(!f5 t/2)] + (l/v'W)cos(t - tan- I(I/3»

(b) Looks like (l/v'W)cos(t - tan- I(l/3». 45. Show that the Wronskian of Y, and Y2 does not

vanish. 47. (a) Subtract two solutions with the same initial

conditions. (b) Show that they are zero when x = o. (c) Solve algebraically for y(x).

49. (a) Compute the derivative of the Wronskian (b) If (a-1)2*4ß and '1"2 are roots, theny

= C,X" + C2X"; if (a - 1)2 = 4ß and , is the root, then y = CIX' + c2x(l-a)/2Inx. (Assume x > 0 in each case).

51. (a) Add alI three forces (b) Substitute and differentiate.

53. cleA + C2eiA + C3e - A + c"e'- jA

where A = (l + i)/,f2 or

ex//i[ blros( ~ ) + b2sin( ~ )]

+ e-x//i [ b3COS( ~ ) + b4sin( ~ ) ] 55. !e X + fex), where fex) is the solution to Exercise

53.

12.8 Serles Solutlons of Differential Equatlons

[00 2n]

1. y = ao ~ 2: 1 + 1-0 n.

a L n. x 2n + 1 [

00 2nl ] I n-O (2n + I)!

[

00 (_I)nx2n+1 ] 3. =a +ax+a

y 0 I I n~' (2n + I)(n + I)!

5. Y = x - x 3/3 + x 5/ 10 - ...

7. y=2x-x3+7x5/20-'"

( X3X6 )

9. Y = ao I + - + - + ... 6 180

( X4 x 7 ) +a, x+-+-+ .. · 12 504

The recursion relation is

an+3 = an/[(n + 3)(n + 2)1

11. Y = C, I + x - - + - - -- + ... + ( X2 x 3 x 4 )

4 60 1920

C2(X4/ 3 _ x 7/ 3 + X '0/ 3 _ x 13/ 3 + ... ) 7 140 5460

or XI / 6 [ blCOS( 11 ~nx ) + b2sin( 11 ~nx )] (no

further terms).

k x k + 2 X k +4

15. (a) x + 4k + 4 + (4k + 4)(8k + 16) + ...

x k + 2j + + ... 4 j (k + 1)(2k +4)··' (jk + /)

-k+2 (b) x- k + ~ 4k + 4

-k+4 + X + ...

(-4k + 4)( -8k + 16)

X- k + 2j + , + ...

4 J( - k + 1)( - 2k + 4) ... ( - jk + /)

17. Solve recursively for roefficients, then rerognize the series for sine and cosine.

19. (a) Use the ratio test (b) x, - t + ~X2, - ~x + !X3

21. Show that the Wronskian is non-zero 23. (a) Solve recursively

(b) Substitute the given function in the equation. (To discover the solution, use the methods for solving first order linear equations given in Section 8.6).

Review Exercises for Chapter 12 1. Converges to 1/11. 3. Converges to 45/2 7. Diverges

11. Converges 15. Diverges 19. Converges

5. Converges to 7/2 9. Converges

13. Converges 17. Diverges 21. Converges

23. Converges 25. 0.78 27. -0.12 29. -0.24 31. 0.25 33. False 35. False 37. False 39. False 41. True 43. True 45. True 47. Use the comparison test. 49. 1/8 51. I 53.R=00 55.R=00 57. R = 2

61. Li"=I[(-I)i+lx 4ili] 63. Lf. 1[( - I )iX 2i I (2i)!] 65. Li"= I[X i I[i(i!)]] 67. Lf_ole2(x - 2Y/i!], R = 00

($J Hf-I)"'O-i+l) . 69. ~ ., (x - I)'

i=O I.

71. '!T2/2 73. 3 75. 3,7,3 -7i,/58

77. ±(I+!ß)I/2~±1.46, +(-I+!l5)1/2~

;:0.344, ~2 + i ~ ± 1.46 ± 0.344i, 4,j5 ~ 1.50 79. z =,fIexp(-'!Ti/4)

y

x

-1

81. z = exp('!Ti) y

x

83. I ±~I - '!Ti ~2.4658 - 1.0717i, - 0.4658 + 1.07 m

85. clcos2x + c2sin2x 87. y = clexp( - 5x) + C2exp( - x) 89. y = - eX 16 - 11 cos xl 130 + 33 sin xl 130 +

clexp( - 5x) + c2exp(2x)

91 3x + ]x + 140 ( x ) 48. (X) • cle C2xe 1369 cos 2" - 1369 sm 2"

ehapter 12 Answers A.67

J xsin(2x) 93. -cos(2x) dx

R+T J xcos(2x)

+ sin(2x) dx R+T

95. CI + e-X(c2cOsx + C3sinx) 97. m = I, k = 9, y = I, Fo = I, g = 2, '" = 3, II

= tan -10). As t ~ 00, the solution approaches

focos[2t - tan-l(~)].

99. m = I, k = 25, y = 6, F= I, g = '!T, '" = 5, II = tan- I [6'!T1(25 - '!T2)]. As t~ 00, the solution apo proaches

I COS['!Tt - tan- I( ~)] J625 - 14'!T2 - '!T4 25 - 772

101. ao( 1- ~3 + ... ) + a l ( x _ ~4 + ... )

103. ao( I _ x2 + x64 _ ~5 + ... )

( x3 x 4 x 5 ) +al x - 3 +"6 - 20 + .. ,

x 2 11x3 105. I - x + T - -6- + ...

x 3 x 5 107. x + "8 + 192 + ...

109. (a) m = L, k = l1 c, y = R (b) O.OI998e-19.90t - O.02020e-O.l00St

+ 0.002099 sin(60xt) + 0.0002227 cos(60xt)

111. Factor out x 2•

113. (a) The partial sums converge to y(x, t) for each (x, t).

(b) Lk-O( -llA 2k + I 115. ~0.659178 117. (a) ~ 1.12

(b) ~ 2.24. It is accurate to within 0.02. 119. -1/2,1/6,0 121. (a) ~ 3.68

(b) y

2

2 x

123. Show by induction that g(n)(x) is a polynomial times g(x).

125. True 127, (a) Collect terms

(b) The radius of convergence is at most I. (c) e

129. Show that a < I1 k by using a Maclaurin series with remainder.

Index Includes Volumes 1 and 11

Note: Pages 1-336 refer to Volume I; pages 337-644 refer to Volume 11.

Abel, Nils Hendrik 172 absolute vaIue 22

funetion 42, 72 properties of 23

absolutely eonvergent 574 accelerating 160 aeeeieration 102, 131

gravitationaI 446 Aehilles and tortoise 568 addition fonnulas, 259 air resistanee 136 Airy's equation 640 aIgebraie operations on power series 591 aIgebraie rules 16 aIternating series test 573 amplitude 372 analytie 600 angular

frequeney 373 rnomentum 506

annuaI percentage rate 382 antiderivative 128

of Ir 323, 342 of eonstant mutiple 130, 338 of exponentiaI 342 of hyperbolie funetions 389 of inverse trigonometrie funetion 341 of lIx 323, 342 of polynomiaI 130 of power 130, 338 mies 337, 338 of sum 130, 338 of trigonometrie funetion 340

of trigonometrie funetions 269 Apostol, Tom M. 582 applieations of the integral, 240 approaches 58 approximation, first-order (see linear

approximation) approximation, linear (see linear

approximation) arc length 477

in polar coordinates 500 Archimedes 3,5,6 area 4,251

between graphs 211, 241 between intersecting graphs 242 in polar coordinates 502 of a sector 252 signed 215 of a surface 482 of a surface of revolution 483 under graph 208, 212, 229 under graph of step funetion 210

argument 40 arithmetie mean 188 arithmetic-geometrie mean inequality 436 astroid 198 astronomy 9 asymptote 165

horizontal 165, 513, 535 vertieal 164, 518, 531

asymptotie 164 average 3

power 464, 465

1.2 Index

average (cont.) rate of change 100 value 434 velocity 50 weighted 437

axes 29 axial symmetry 423 axis of symmetry 440

8·1) definition of limit 516 ball 421 Bascom, Willard 306(fn) base of logarithm 313 beats 628 Beckman, P. 251, fn Berkeley, Bishop George 6(fn) Bemoulli, J. 252(fn), 521

equation 414 numbers 643

Bessel, F.W. 639 equation 639 functions 643

binomial series 600 bisection, method of 142, 145 blows up 399 bouncing ball 549 bounded above 575 Boyce, William 401 Boyer, C. 7(fn), 252(fn) Braun, Martin 380,401, 414, 426 Burton, Robert 8 bus, motion of 49, 202, 207, 225

Calculator discussion 49, 112, 166, 255, 257, 265, 277, 309, 327, 330, 541

calculator symbol 29 calculus

differential 1 fundamental theorem of 4, 225, 237 integral 1,3

Calculus Unlirnited iii, 7(fn) capacitor equation 406 carbon-14 383 ClU'dano, Girolamo 172 cardioid 298 cartesian coordinates 255 catastrophe

cusp 176 theory 176

catenary 402 Cauchy, Augustin-Louis 6, 521

mean value theorem 526 Cavalieri, Bonaventura 8,425 center of mass 437

in the plane 439 of region under graph 441

of triangular region 445 chain rule 112

physical model 116 change

average rate of 100 instantaneous rate of 10 linear or proportional 100 proportional 95 rate of 2, 100, 101, 247 of sign 146 total 244

chaos, in Newton's method 547 characteristic equation 617 chemical reaction rates 407 circle 34,44, 120,251,421

equations of 37 parametric equations of 490

circuit, electric 413 circular functions 385 circumference 251 city

Fat 116 Thin 115

climate 180 closed interval 21

test 181 College, George 383 common sense 61, 193 comparison test 570

for improper integrals 530 for limits 518 for sequences 543

completing the square 16, 17, 463 complex number 607, 609

argument of 611 conjugate of 611 imaginary part of 611 length or absolute value of 611 polar representation of 614 properties of 612 real part of 611

composition of functions 112, 113 derivative of 113

concave downward 158 upward 158

concavity, second derivative test for 159 conditionally convergent 574 conoid 486 conservation of energy 372 consolidation principle 438 constant function 41, 192

derivative of 54 rule for limits 62, 511

constant multiple rule for antiderivatives 130 for derivatives 77 for limits 62

for series 566 eonsumer's surplus 248 eontinuity 63, 72

of rational funetions 140 eontinuous 139 eontinuous flmetion 63

integrability of 219 continuously compounded interest 331,

382, 416 convergence, absolute 574

conditional 574 of series 562 of Taylor series 597 radius of 587

eonvergent integral 529 convex funetion 199 eooling, Newton's law of 378 coordinates 29

eartesian 255 polar 253, 255

eoriolis force 499 eosecant 256

inverse 285 eosine 254

derivative of 266 hyperbolic 385 inverse 283 series for 600

eosines, law of 258 cost, marginal 106 cotangent 256

inverse 285 Creese, T.M. 401 critical points 151 critically damped 621 cubic function 168

general, roots and graphing 172 curve 31(fn)

parametrie 124, 298, 489 cusp 170

catastrophe 176 eycloid 497

dam 454 damped force oseillations 628 damping 377

in simple harmonie motion 415 Davis, Phillip 550 day

length of 30, 302 shortening of 303

decay 378 decimal approximations 538 declerating 160 decrease, rate of 101 deereasing function 146 definite integral 232

Index 1.3

by substitution 355 eonstant multiple ruIe for 339 endpoint additivity rule for 339 inequality rule 339 power rule 339 properties of 234, 339 sum rule 339 wrong-way 339

degree as angular measure 252 of polynomial and rational functions 97

delicatessen, Cavalieri's 425 delta 50(fn) demand eurve 248 Demoivre, Abraham 614

formula 614 density 440

uniform 440 depreciation 109 derivative 3, 53, 70

of Ir 318 of eomposition of funetions 113 of constant multiple 77 of eosine 266 formal definition of 70 of hyperbolie funetions 388 of implicitly defined funetion 122 of integer power 87 of integral with respect to endpoint 236 of integral whose endpoint is a given

function 236 of inverse hyperbolic funetions 396 of inverse funetion 278 of inverse trigonometrie funetions 285 Leibniz notation for 73 as a limit 69 of linear funetion 54 logarithmie 117, 322, 329 of logarithmie funetion 321 of lIx 71 of polynomial 75, 79 of power 75, 119 of power of a function 110, 119 of produet 82 of quadratie function 54 of quotient 85 of rational power 119 of rational power of a funetion 119 of reciprocal 85 of sum 78 second 99, 104, 157 summary of rules 88 of Vx 71

Dido 182 differenee quotient 53 differentiable 70 differential

algebra 356

1.4 Index

differential (cont.) ealeulus 1 equation 369

Airy's 369 Bessel' s 639 first order 369 harmonie oseillator 370 Hermite' s 636 Legendre's 635 linear first order 369 numerieal methods for 405 of growth and decay 379 of motion 369 seeond order linear 617 separable 398, 399 series solutions 632 solution of 39 spring 370 Tehebyehefrs, 640

differential notation 351, 359, 374, 398 differentiation 3, 53, 122, 201

implicit 120, 398 logarithmie 117, 322, 329 of power series 590

diminishing returns, law of 106 Diprima, Riehard 390, 401 direetion field 403 diseriminant 17 disk 421

method 423 displaeement 230 distanee formula, in the plane 30

on the line 23 divergent integral 529 domain 41 double-angle formulas 259 drag 136

resistanee 414 dummy index, 203

e, 319, 325 as a limit, 330

E-A definition of limit 513 E-8 definition of limit 509 ear popping 116 earth's axis, inelination of 301 economies 105 electrie eireuit 399, 413 element 21 elliptie integral 417, 506, 507 endpoints 181

of integration 17 energy 201,445

conservation of 372 potential 446

equation of circle and parabola 37

differential (see also differential equation)

pararnetric 124, 298 simultaneous, 37 spring 376 of straight line 32 of tangent line 90

error function 558 Eudoxus 4 Euler, Leonhard 251(fn), 252(fn), 369

method 404 equation 636 formula 608

evaluating 40 even function 164, 175 exhaustion method of 5, 7 existence theorem 180, 219 exponent, zero 23 exponential

function 307 derivative of 320 graphing problems 326 limiting behavior 328

growth 332 series 600 spiral 310, 333

exponentiation 23 exponents

integer 23 laws of 25 negative 26 rational 27, 118 real, 308

extended product rule for limits 62 extended sum rule for limits .62, 69

extensive quantity 445 extreme value theorem 180

factoring 16 falling object 412, 414 Feigenbaum, Mitchell J. 548 Ferguson, Helaman 602 Fermat, Pierre de 8 Fine, H.B. 468 first derivative test 153 first-order aproximation (see linear

approximation) ftying saucer 430 focusing property of parabolas 36, 95, 97 football 453 force 448

on a darn 454 forced oscillations 415, 626 Fourier coefficients 506 fractals 499 fractional (see rational) frequeney 259

friction 377 Frobenius, George 636 frustum 485 funetion 1, 39

absolute value 42, 72, 73 average value of 434 eomposition of 112, 113 eonstant 41, 192 continuous 63 eonvex 199 eubie 168 definition of 41 differentiation of 268 even 164, 175 exponential 307 graph of 41, 44 greatest integer 224 hyperbolic 384, 385 identity 40, 277, 384, 385 inverse 272, 274 inverse hyperbolie 392 inverse trigonometrie 281, 285 linear 192 odd 164,175 piecewise linear 480 power 307 rational 63 squaring 41 step 140, 209, 210 trigonometrie, antiderivative of 269 trigonometrie, graph of 260 zero 41

fundamental integration method 226 set 630

fundamental theorem of ealeulus 4, 225, 237

alternative version 236

Galileo 8 gamma funetion 643 Gauss, Carl Friedrieh 205, 615 Gear, Charles W. 405 Gelbaum, Bernard R. 576, 600 general solution 618, 623 geometrie mean 188, 436

series 564,600 global 141, 177 Goldstein Larry 172 Gould, S.H. 6(fn) graphing in polar coordinates 296 graphing problems

exponential and logarithmie funetions 326

trigonometrie funetions 292 graphing procedure 163 graphs 41, 163

area between 241

area under 212, 229 of funetions 41, 44

gravitational acceleration 446 greatest integer funetion 224 growth 378

Index 1.5

and deeay equation, solution 379 exponential 332

half-life 381, 383 hanging eable 401 Haraliek, R.M. 401 Hardin, Garrett 416 harmonie series 567 Hermite polynomial 636 Hermite's equation 636 herring 156 Hipparchus 256(fn) Hofstadter , Douglas 548 Hölder eondition 559 homogenous equation 623 Hooke's Law 99,295 horizontal asymptote 165, 513, 535 horizontal tangent 193 horsepower 446 horseraee theorem 193 hyperbolie eosine 385 hyperbolie funetions 384, 385

derivatives 388 antiderivatives 389 inverse 392

hyperbolie sine 385 inverse 393

I method 361 identity funetion 40, 277

rule for limits 60 identity, trigonometrie 257 illumination 183 imaginary axis 609 imaginary numbers 18 implicit differentiation 120, 122, 398 improper integrals 528, 529

comparison test for 530 inelination of the earth' s axis 30 1 inerease, rate of 101 increasing funetion 146

test 148 theorem 195 sequence property 575

inereasing on an interval 149 indefinite integral (see antiderivative) indefinite integral test 233 independent variable 40 indeterminate form 521 index

dummy 203

1.6 Index

index (cont.) substitution of 205

indicial equation 638 induetion, principle of 69 inequality 18

arithmetie-geometrie mean 188, 436 Minkowski' s 365 properties of 19

infinite limit 66 infinite series 561 infinite sum 561 infinitesimals 73

method of 6, 8 infinity 21 infleetion point 159

test for 160 initial eonditions 371, 398 instantaneous quantity 445 instantaneous velocity 50, 51 integer power rule for derivatives 87 integers 15

sum of the first n 204 mtegrability of eontinuous funetion 219 intehable 217 integral 217

ealeulated "by hand" 212 ealeulus 1 eonvergent 529 definite 232 definition of 217 divergent 529 elliptie 417 of hyperbolie funetion indefinite 129 (see also antiderivative) improper 528, 529 of inverse funetion 362 Leibniz notation for 132 mean value theorem 239 mean value theorem 435 of rational funetions 469 of rational expression in sin x and eos

x 475 Riemann 220 sign 129, 132, 217 tables 356 trigonometrie 457, 458 of unbounded funetions 531 wrong way 235

integrand 129 integration 33, 129, 201

applieations of 420 by parts 358 by substitution 347, 348, 352 endpoint of 217 limit of 217 method, fundamental 226 methods of 337 numerieal 550

of power series 590 intensity of sunshine 451 interest, compound 244, 331 intermediate value theorem 141, 142 intersecting graphs, area between 242 intersection points 39 interval 21

elosed 21 open 19

inverse eosecant 285 eosine 283 eotangent 285 funetion 272, 274

integral of 362 rule 278 test 276

hyperbolie funetions 392 integrals 396 derivatives 396

hyperbolie sine 393 secant 285 sine 281 tangent 283 trigonometrie funetions 281, 285

invertibility, test for 275, 276 irrational numbers 16 ith term test 567

joule 445

Kadanoff, Leo 548 Keisler, H. Jerome 7(fn), 73(fn) Kelvin, Lord 594 Kendrew, W.G. 180 Kepler, Johannes 8

second law 506 Kilowatt-hour 446 kinetie energy 446 Kline, Morris 182

I'Höpital, Guillaume 521 rule 522, 523, 525

labor 106 ladder 190 Lagrange's interpolation polynomial 556 Laguerre funetions 640 Lambert, Johann Heinrich 251(fn) latitude 300 law of mass action 476 law of refleetion 290 Legendre, Adrien Marie 251(fn)

equation 635 polynomial 635

Leibniz, Gottfried 3, 73, 193(fn), 594

notation 73, 104, 132, 217 for derivative 73 for integral 132

lemniscate 136 length

of curves 477 of days 300, 302 of parametrie eurve 495

librations 506 limac;on 298 limit 6, 57, 59

at infinity 65, 512 eomparison test 518 of (eos x - 1)/x 265 derivative as a 69 derived properties of 62 e-& definition of 509 of function '509 infinite 66 of integration 217 method 6 one-sided 65, 517 of powers 542 product rule 511 properties of 60, 511 reeiproeal rule 511 of sequence 537, 540

properties 563 of (sin x)/x 265

line 31(fn) equation of 32 perpendicular 33 point-point form 32 point-slope form 32 real number 18 secant 51, 191 slope of 52 slope-intereept form 32 straight 31(fn), 125 tangent 2, 191

linear approximation 90,91,92, 158, 159, 601

linear funetion 192 derivative of 54

linear or proportional change 100 linearized oseillations 375 Lipshitz condition 559 Lissajous figure 507 loeal 141, 151, 177

maximum point 151, 157 minimum point 151, 157

logarithm 313 base of 313 defined as integral 326 function, derivative of 321 laws of 314 limiting behavior 328 natural 319

properties of 314 series for 600 word problems for 326

Index 1.7

logarithmie differentiation 117,322,329 logarithmie spiral 534, 535 logistic equation 506 logistic law 407 logistie model for population 335 Lotka-Voltera model 400 love bugs 535 lower sum 210 Luean 8(fn)

Maclaurin, Colin 594 polynomial for sin x 602 series 594, 596

MACSYMA 465 majorize 199 Mandelbrot, Benoit 499 marginal

cost, 106 produetivity 106 profit 106 revenue 106

Marsden, Jerrold 582, 615 Matsuoka, Y. 582 maxima and minima, tests for 153, 157

181 '

maximum global 177 point 151 value 177

maximum-minimum problems 177 mean value theorem 191

Cauchy's 526 eonsequences of 192 for integrals 239, 435, 455

Meech, L.W. 9 midnight SUD 30l(fn) minimum

points 177 value 177

Minkowski's inequality 365 mixing problem 413, 414 modulates 628 motion, simple harmonic 373

with damping 415

natural growth or decay 380 logarithms 319 numbers 15

Newton, Isaac, 3(fn), 8(fn), 193(fn), 253(fn), 594

iteration 559 law of eooling 378 method 537, 546

1.8 Index

Newton, Isaac (cant.) accuracy of 559 and chaos 547

second law of motion 369 nonhomogenous equation 623 noon 301(fn) northem hemisphere 301 notation

differential 351,359,374,398 Leibniz 73, 104, 132, 217 summation 203, 204

nowhere differentiable continuous function 578

number complex 607, 609 imaginary 18 irrational 16 natural 15 rational 15 real 15, 16

numerical integration 550

odd function 164, 175 Olmsted, John M. H. 578, 600 one-sided limit 65, 517 open interval 21 optical focusing property of parabolas 36,

95, 97 order 18 orientation quizzes 13 origin 29 orthogonal trajectories 402 oscillations 294, 369

damped forced 628 forced 415 harmonic 373 linearized 375 overdamped 621 underdamped 621

oscillator, forced 626 oscillatory part 629 Osgood, W. 521 overdamped case 621

pH 317 Papprus theorem 454 parabola 34

equations of 37 focusing property of 36, 95, 97 vertex of 55

parameter 489 parametric curve 124,287,489

length of 495 tangent line to 491,492

parametric equations of circle 490 of line 490

partial fractions 465, 469, 591 partial integration (see integration by parts) particular solution 371, 623 partition 209 parts, integration by 358, 359 pendulum 376,391,417 period 259 periodic 259 perpendicular lines 33 Perverse, Arthur 367 pharaohs 416 phase shift 372, 629 Picard's method 559 plotting 29, 43, 163 point

critical 151 inflection 159 interseetion 39 local maximum 151, 157 local minimum 151, 157

point-point form 32 polar coordinates 253, 255

arc length in 500 area in 502 graphing in 296 tangents in 299

polar representation of complex numbers 611

Polya, George 182 polynomial

antiderivative of 130 derivative of 75, 79

pond, 74 population 117, 175, 189, 195, 335, 344,

382, 400, 407, 416 position 131 Poston, Tim 176 potential energy 446 power 445

function 307 integer 23 negative 26 of function rule for derivatives 110 rational 18, 27, 169 real 308 rule

for antiderivatives 130 for derivatives 76, 119 for limits, 62

series 586 algebraic operations on 591 differentiation and integration of 590 root test 589

predator-prey equations 400 producer's surplus 248 product ruIe

for derivatives 82

product rule (cont.) for limits 60

E-8 proof 520 productivity

of labor 106 marginal 106

profit 329 marginal 106

program 40 projectile 295 proportional change 95 Ptolemy 256(fn) pursuit curve 499 Pythagoras, theorem of 30

quadratic formula 16, 17 function

derivative of 54 general, graphing of 176

quizzes, orientation 13 quotient

derivative of 85 difference 53 rule, for limits 62

radian 252 radius 34

of convergence 587 rate of change 2, 101, 247

decrease 101 increase 10 1 relative 329

rates, related 124 ratio comparison test for series 571 ratio test

for power series 587 for series 582

rational exponents 118 expressions 475 function, continuity of 63, 140 numbers 15 power rule, for derivatives of a

function 119 powers 118, 119

rationalizing 28 substitution 474

real axis 609 real

exponents 308 number line 18 numbers 15, 16 powers 308

reciprocal rule for derivatives 86

for limits 60 test for infinite limit 517

recursively 541 reduction formula 365 reduction of order 619 reflection, law of 290 region between graphs 240 related rates 124

word problems 125 relative rate of change 329 relativity 80(fn)

Index 1.9

repeated roots 620 replacement rule, for limits 60 resisting medium 412 resonance 415, 626, 629 revenue, marginal 106 revolution, surface of 482 Riccati equation 414 Richter scale 317 Riemann, Bernhard 220, fn

integral 220 sums 220, 221, 551

Rivlin's equation 199 Robinson, Abraham 7, 73(fn) rocket propulsion 412 Rodrigues' formula 640 Rolle, Michel 193(fn)

theorem 193 root splitting 619 root test

for power series 589 for series 584

rose 297 Ruelle, David 548 Ruffini, Paolo 172

Saari, Donald G. 548 scaling rule, for integral 350 Schelin, Charles W. 257 (fn) school year 303 secant, 256

inverse 285 line 52, 191

second derivative 99, 104, 157 test for maxirna and minirna 157 test for concavity, 159

second-order approximation 601 second-order linear differential

equations 617 sector, area of 252 separable differential equations 398, 399 sequence 537

comparison test 543 limit of 537, 540 properties 563

series altemating 572

1.10 Index

series (cant.) comparison test for 570 constant multiple rule for 566 convergence of 562 divergent 562 geometrie 564 harmonie 567 infinite 561 integral test 580 P 581 power (see power series) ratio comparison test for 571 ratio test for 582 ooot test for 584 solutions 632 sum of 562 sum rule for 566

set 21 shell method 429 shifting rule

for derivatives 115 for integral 350

sigma 203 sign, change of 146 signed area 215 similar triangles 254 Simmons, George F. 401 simple juumonic motion 373

damped 415 Simpson's rule 554 simultaneous equations 37 sine 254

derivative of 266 hyperbolic 385 inverse 281 series 600

sines, law of 263 slice method 420 slope 2, 31

of tangent line 52 slope-intercept form 32 Smith, D.E. 193(fn) Snell's law 305 solar energy 8, 107, 179, 180,221,449 solids of revolution 423,429 Spearman-Brown formula 520 speed 103, 497 speedometer 95 sphere 421

bands on 483 spiral

exponential 310, 333 logarithmic 534, 535

Spivak, Mike 251(fn) spring

constant 370 equation 370, 376

square, completing the 16, 17, 463

square ooot function, continuity of 64 squaring function 41 stable equilibrium 376 standard deviation 453 steady-state current 520 step function 5, 140, 209, 210

area under graph 210 straight line 31(fn), 125 (see also line) stretching rule, for derivatives 117 strict local minimum 151 Stuart, Ian 176 substitution

definite integral by 355 integration by 347, 348, 352 of index 205 rationalizing 474 trigonometrie 461

sum rule 8-8 proof 520 for derivatives 78 for limits 60 physical model for 80

sum 203 collapsing 206 infinite 561 lower 210 of the first n integers 204 Riemann 220,221,551 rule for antiderivatives 130 telescoping 206 upper 210

summation notation 201, 203, 204 properties of 204, 208

sun 300 sunshine intensity 451 superposition 371 supply curve 248 surface of revolution 482

area of 483 suspension bridge 407 symmetries 163, 296 symmetry

axis of 440 principle 440

tables of integrals 356, endpapers Tacoma bridge disaster 626 tangent

hyperbolic 386 inverse 284 line 2, 191,491

horizontal 193 slope of 52 to parametrie curve 492 vertical 169

tangentfunction 256

Tartaglia, Nieeolo 172 Taylor series 594

test 59 eonvergenee of 597

Taylor, Brook 594 Tehebyeheff's equation 640 teleseoping sum 206 terminal speed 412 third derivative test 160 Thompson, D'Arcy 423 time

of day 301 of year 301

torus 431 total ehange 244 traetrix 499 train 55, 80, 291 transeontinental railroad 569 transient 411, 628 transitional spiral 643 trapezoidal rule 552 triangles , similar 254 trigonometrie funetions

antiderivatives of 269 derivatives of 264, 268 graphing problems 282 inverse 281, 285 word problems 289

trigonometrie identity 257 trigonometrie integrals 457, 458 trigonometrie substitution 461 triseeting angles 172

unbounded region 528 underdamped oseillations 621 undetermined eoefficients 623 unieellular organisms 423 uniform density 440 uniform growth or decay 381 unstable equilibrium 376, 390, 406 upper sum 210 Urenko, John B. 548

value absolute (see absolute value) maximum 177 minimum 177

variable ehanging 354

independent 40 varianee 453

Index 1.11

variation of eonstants (or parameters) 378, 624

velocity 102, 131, 230 average 50 field 404 instantaneous 50, 51 positive 149

vertex 55 vertieal asymptote 164, 518, 531 vertieal tangent 169 Viete, Fran~ois 251(fn) Volterra, Vito 401 volume

of bologna 426 disk method for 423 shell method for 429 sliee method for 419 of a solid region 419 washer method for 424

washer method 424 water 178, 247

fiowing 131, 144,343 in tank 126

watt 446 wavelength 263 waves, water 306 Weber-Feehner law 33 Weierstrass, Karl 6, 578 weighted average 437 window seat 291 word problems

integration 247 logarithmie and exponential

funetions 326 maximum-minimum 177 related rates 125 trigonometrie funetions 289

wrong-way integrals 235 Wronskians 630

yogurt 279

zero exponent 23 funetion 41

Undergraduate Texts in Mathematics (continued from page ii)

HiitonIHoltonIPedersen: Mathematieal Refleetions: In a Room with Many Mirrors.

Iooss/Joseph: Elementary Stability and Bifureation Theory. Seeond edition.

Isaac: The Pleasures ofProbability. Readings in Mathematics.

James: Topologieal and Uniform Spaees.

Jänich: Linear Algebra. Jänich: Topology. Kemeny/Snell: Finite Markov Chains. Kinsey: Topology of Surfaees. KIambauer: Aspeets ofCalculus. Lang: A First Course in Calculus. Fifth

edition. Lang: Caleulus ofSeveral Variables.

Third edition. Lang: Introduetion to Linear Algebra.

Seeond edition. Lang: Linear Algebra. Third edition. Lang: Undergraduate Algebra. Seeond

edition. Lang: Undergraduate Analysis. Lax/BursteinlLax: Calculus with

Applieations and Computing. Volume 1.

LeCuyer: College Mathematics with APL.

LidIlPilz: Applied Abstract Algebra. Second edition.

Logan: Applied Partial Differential Equations.

Macki-Strauss: Introduction to Optimal Control Theory.

Malitz: Introduction to Mathematical Logic.

Marsden/Weinstein: Calculus I, 11, III. Second edition.

Martin: The F oundations of Geometry and the Non-Euclidean Plane.

Martin: Geometrie Constructions. Martin: Transformation Geometry: An

Introduction to Symmetry. Millman/Parker: Geometry: AMetrie

Approach with Models. Second edition.

Moschovakis: Notes on Set Theory.

Owen: A First Course in the Mathematical Foundations of Thermodynamies.

Palka: An Introduction to Complex Function Theory.

Pedrick: A First Course in Analysis. PeressinilSullivanlUhl: Tbe Mathematics

ofNonlinear Programming. PrenowitzlJantosciak: Join Geometries. Priestley: Calculus: A Liberal Art.

Second edition. ProtterlMorrey: A First Course in Real

Analysis. Second edition. ProtterlMorrey: Intermediate Calculus.

Second edition. Roman: An Introduction to Coding and

Information Theory. Ross: Elementary Analysis: The Theory

of Calculus. Samuel: Projective Geometry.

Readings in Mathematics. ScharlaulOpolka: From Fermat to

Minkowski. Schiff: The Laplace Transform: Theory

and Applications. Sethuraman: Rings, Fields, and Vector

Spaces: An Approach to Geometrie Constructability.

Sigler: Algebra. SilvermanlTate: Rational Points on

Elliptic Curves. Simmonds: A Brief on Tensor Analysis.

Second edition. Singer: Geometry: Plane and Faney. Singerrrhorpe: Lecture Notes on

Elementary Topology and Geometry.

Smith: Linear Algebra. Third edition. Smith: Primer ofModem Analysis.

Seeond edition. Stanton/White: Construetive

Combinatories. StillwrlI: Elements of Algebra:

Geometry, Numbers, Equations. StilIweIl: Mathematies and lts History. StilIweIl: Numbers and Geometry.

Readings in Mathematics. Strayer: Linear Programming and Its

Applieations.

Undergraduate Texts in Mathematics

Thorpe: Elementary Topics in Differential Geometry. Toth: Glimpses of Algebra and Geometry.

Readings in Mathematics. Troutman: Variational Calculus and

Optimal Contro!. Second edition.

Valenza: Linear Algebra: An Introduction to Abstract Mathematics.

Whyburn/Duda: Dynamic Topology. Wilson: Much Ado About Calculus.

ABrief Table of Integrals, continued.

29. J cschx dx = Inltanh 11 = - i In ~~:~~ ~ ! 30. J sinh2x dx = ± sinh 2x - i x

31. J cosh2x dx = ± sinh 2x + i x

32. J sech2x dx = tanh x

33. JSinh -I ~ dx = x sinh -I ~ -~ a a

(a > 0)

{XCOSh-l~ -~ 34. J cosh - 1 ~ dx = a a xcoSh-l~ +~

a

[ cosh - 1 ( ~ ) > 0, a > 0 J

[ cosh -I( ~ ) < 0, a > 0] 35. J tanh -I ~ dx = x tanh -I ~ + ~ Inla2 - x21

36. J I dx = In(x + ~a2 + x 2 ) = sinh - 1 ~ ~a2 + x2 a

(a > 0)

37. J __ 1_ dx = ! tan - I ~ (a > 0) a2 + x 2 a a

J~ x~ a2 x 38. va -x dX=Iva -x +Tsin-'a (a>O)

39. J (a 2 - x 2)3/2 dx = i (5a 2 - 2X2)~ + 3t sin -I ~

40.J I dx=sin-I~ (a>O) ~a2 _ x2 a

41. J __ 1_ dx = ~ Inl a + x 1 a2 _ x 2 2a a - x

42. J 1 3/2 dx = x (a 2 - x 2) a2~a2 - x 2

43. J ~X2 ± a2 dx = I ~x2 ± a2 ± ;2 Inlx + ~x2 ± a2 1

44. J I dx = Inlx +~I = cosh- I ~ ~ a

45. J I dx = ! Inl-x-I x (a + bx) a a + bx

J r;::-;-;:::: 2(3bx - 2a)(a + bx)3/2

46. xya + bx dx = ------=----15b 2

47. J 7 dx = 2..fa + bx + aJ 1 x,Ja + bx

2(bx - 2a),Ja + bx 48. J x dx = -------;:----

,Ja + bx 3b2

dx

(a > 0)

49. ( 1 dx = ~ Inl ,;a+iiX -..;a I .J x,Ja + bx ..;a,Ja + bx +..;a

(a > 0)

= _2_ tan -I ~ a + bx ~ -a

(a < 0)

50. J ~ dx =~a2 - x 2 -alnl a +~ I 51. f x~ dx = - j (a2 - x 2)3/2

(a > 0)

(a > 0)

Continued on overleaf.

ABrief Table of Integrals, continued.

53. J 1 dx = - .!.lnj a + ~ I x~ a x

54.J x dx=-~ Ja 2 - x 2

J X2 X ~ a2 x 55. dx = - 2' Va2 - x + 2 sin -I a ~

(a > 0)

56.J~ dx=R+?-alnla+~1 57. J ~ dx=~ -acos- I I: I

=Jx2 -a2 -asec-I(~) (a>O)

58. J xJx2 ± a2 dx = j (x 2 ± a2)3/2

59. J 1 dx = .!.lnl x I xR+? a a+R+?

60.J 1 dx=.!.COS-I~ (a>O) x~ a lxi

61 J 1 dx= +-~ • 2 X 2Jx 2 ± a2 a x

62. J x dx = Jx 2 ± a2 Jx 2 ± a2

63 J 1 dx = 1 In!2ax + b - Jb 2 - 4ac! (b 2 > 4ac) . ax2 + bx + C Jb2 - 4ac 2ax + b + Vb2 - 4ac

2 tan-I 2ax + b (b2 < 4ac) J4ac - b2 v4ac - b2

64. J x dx = -21 Inlax2 + bx + cl - 2b J 1 dx ax2 + bx + c a a ax2 + bx + c

65. J 1 dx = ~ Inl2ax + b + 2/ii vax 2 + bx + cl (a > 0) vax2 + bx + c /ii

= _I _ sin - 1 - 2ax - b (a < 0)

r:::a Vb 2 - 4ac

66. J vax2 + bx + c dx = 2a~ + b vax2 + bx + c + 4acS - b2 J 1 dx a a vax2 + b + c

67. J x dx = vax2 + bx + c -..!!...- J 1 dx Jax 2 + bx + c a 2a Jax 2 + bx + c

68. J 1 dx = ..::...!.lnI 2..[c Jax 2 + bx + c + bx + 2c I (c > 0) xvax2 + bx + c..[c x

=_l-sin- I bx+2c (c<O) Fe Ixlvb2 - 4ac

69. J x3,~ dx =( ~x2 - -lsa2)~(a2 + X 2)3

J Jx2 ± a2 +- ~(x2 ± a2)3

70. dx = --'------:--::---x 4 3a2x 3

J .. sin(a - b)x sin(a + b)x

71. smaxsmbxdx= 2(a-b) - 2(a+b)

Continued on inside back cover.

ABrief Table of Integrals, continued.

f cos(a - b)x cos(a + b)x

72. sin ax cos bx dx = - 2(a _ b) 2(a + b) sin(a + b)x

+ ---,,-:--~-2(a+ b)

f sin(a - b)x

73. cosaxcosbxdx = 2(a _ b)

74. f sec x tanx dx = secx

75. fcscxcotxdx= -cscx

76. cosmx smnx dx = + --- cosm- 2x sinnx dx f . cosm-Ix sinn + IX m I f m+n m+n

. n-I m+1 I = _ sm x cos x + ~ fcosmx sinn- 2x dx

m+n m+n

77. f xnsinaxdx = - ~ xncosax + ~ f xn-'cosaxdx

78. f xncosaxdx = ~ xnsinax - ~ f xn-Isinaxdx

79. f xne ox dx = x n: ox - ~ f xn-Ieox dx

80. fxnlnaxdx = x n+ l [ Inax _ I ] n + I (n + 1)2

f xn+1 m f 81. xn(lnax)mdx = n + I (lnax)m - n + I xn(lnaxr- 1 dx

f eOX(asinbx - bcosbx)

82. eoxsinbxdx = ------:::----:::--­a2 + b2

f eOX(bsinbx + acosbx)

83. eOXcos bx dx = ---------,--­a2 + b2

84. f sech x tanh x dx' = - sech x

85. f csch x coth x dx = - csch x

Greek Alphabet

a alpha iota ß beta IC kappa y gamma A lambda 8 delta /L mu t: epsilon p nu r zeta ~ xi '1/ eta 0 omicron (J theta 'Tr pi

p rho 0 sigma T tau v upsilon

cf> phi X chi

'" psi

w omega