Exercises in Mathematical Analysis I - uniroma2.itciolli/ExAn1.pdf · Università di Tor Vergata – Dipartimento di Ingegneria Civile ed Ingegneria Informatica Exercises in Mathematical

Università di Tor Vergata – Dipartimento di Ingegneria Civile ed Ingegneria Informatica

Exercises in Mathematical Analysis I

Alberto Berretti, Fabio Ciolli

2

1 Fundamentals

1.1 Polynomial inequalities

Solve the following inequalities for x ∈ R:

Ex. 1. (x3− 3x + 2)(x − 4) > 0.

[x < −2, x > 4

]Ex. 2. (1 − x)(x − 3)(x + 2) < 0.

[− 2 < x < 1, x > 3

]1.2 Rational inequalities


Ex. 3.x2 + x − 2

x2 − 10x + 21<

x − 1x − 3

+ 3x + 1x − 7

.[x < 0, 3 < x < 5, x > 7

]Ex. 4.

x + 12x + 8

−x − 6

x2 + 2x − 48≥

3x − 3x − 6

.[− 8 < x < 6

]Ex. 5.

−9x2− 12x − 4

2x2 − 5x + 2< 0.

[x < −

23, −

23< x <

12, x > 2

]Ex. 6.

(x − a)(x − b)x2 − a2 ≥ 0, a > b > 0.

[x < −a, b ≤ x < a, x > a

]1.3 Irrational inequalities


Ex. 7. 2x − 3 >√

4x2 − 13x + 3.[x ≥ 3

]Ex. 8. x − 8 <

√

x2 − 9x + 14.[x ≤ 2, x ≥ 7

]Ex. 9.

√x − 1 −

√x − 2 < 2.

[x ≥ 2

]Ex. 10.

√x + 2 < 8 +

√x − 6.

[x ≥ 6

]Ex. 11.

√3x − 8 >

√5x + 3 +

√x + 6.

[no solution

]Ex. 12.

√x − 1 ≤ x − 2.

[x ≥

5 +√

52

]

3

A. Berretti, F. Ciolli Exercises inMathematical Analysis I

Ex. 13.√

x − 1 ≥ −100 − x.[x ≥ 1

]Ex. 14.

√x − 2√

x − 4< 1.

[2 ≤ x < 16

]Ex. 15. 3√

|x + 8| > 1.[x < −9, x > −7

]Ex. 16.

√4 − |x + 3| < 2.

[− 7 ≤ x ≤ 1

]Ex. 17. 3√4 − |x + 3| < 2.

[R]

Ex. 18.√

4 − |x + 2| < 2 − |x|.[−5 +

√17

2< x < 1

]Ex. 19.

√3 − |4x + 2| < 1 − 2|x|.

[0 < x ≤

14

]1.4 Absolute value inequalities


Ex. 20. | |x − 1| − 1 | ≥ 2.[{x ≤ −2} ∪ {x ≥ 4}

]Ex. 21. |x − 2| − |x| < 3.

[R]

Ex. 22. | |x − 2| − |x| | ≤ 3.[R]

Ex. 23. |x2− 2x − 4| ≥ |x| + 2.

[ {x ≤ −2} ∪ {x ≥

3 +√

332

}∪

{3 −√

172

≤ x ≤ 2} ]

Ex. 24. |x − 2| + |x| < 3.[ {−

12< x <

52

} ]Ex. 25.

∣∣∣∣∣x − 2x − 3

∣∣∣∣∣ − |x − 2| < 2.[{x < 1 +

√3} ∪ {x > 2 +

√2}

]1.5 Exponential and logarithmic inequalities


Ex. 26. 4x+163x−2 < 8x.[x <

2 log 3log 108

]Ex. 27. 3 · 52(2x−7)

− 4 · 5(2x−7) + 1 > 0.[x <

72−

log 32 log 5

, x >72

]Ex. 28. log3(2x2

− 7x + 103) > 2.[R]

4


Ex. 29. log5(x2− 7x + 11) < 0.

[2 < x <

7 −√

52

,7 +√

52

< x < 5]

Ex. 30. log10(x + 4)2 > log10(13x + 10).[−

1013< x < 2, x > 3

]Ex. 31. 22x

− 5 · 2x + 4 < 0.[0 < x < 2

]Ex. 32.

62x − 1

+3

2x + 1>

22x − 1

+ 5.[0 < x < 1

]Ex. 33. | log10(3x + 4) − log10 7| < 1.

[−

1110< x < 22

]1.6 Trigonometric inequalities


Ex. 34. 2 sin2 x − cos x − 1 > 0.[π

3+ 2kπ < x < π + 2kπ, π + 2kπ < x <

53π + 2kπ, k ∈ Z

]Ex. 35. cos 2x + 3 sin x ≥ 2.

[π6

+ 2kπ ≤ x ≤56π + 2kπ, k ∈ Z

]Ex. 36. 3 tan2 x − 4

√3 tan x + 3 > 0.

[−π2

+ kπ < x <π6

+ kπ,π3

+ kπ < x <π2

+ kπ, k ∈ Z]

Ex. 37. loga

(12− | sin x|

)< 0, a > 1.[

−16π + 2kπ < x <

16π + 2kπ,

56π + 2kπ < x <

76π + 2kπ, k ∈ Z

]Ex. 38. 3 cos x + sin2 x − 3 > 0.

[not possible

]Ex. 39. 4 cos

(x +

π6

)− 2√

3 cos x + 1 ≥ 0.[−

76π + 2kπ ≤ x ≤

π6

+ 2kπ, k ∈ Z]

Ex. 40.∣∣∣∣∣cos 2x

sin x

∣∣∣∣∣ ≤ 1.[π

6+ 2kπ ≤ x ≤

56π + 2kπ,

76π + 2kπ ≤ x ≤

116π + 2kπ, k ∈ Z

]Ex. 41.

∣∣∣∣∣ tan 2xcot x

∣∣∣∣∣ < 1.[kπ < x <

π6

+ kπ,56π + kπ < x < π + kπ, k ∈ Z

]1.7 Boundedness of numerical sets

Study the boundedness of the following numerical sets, expressing for any of them sup, inf, max

and min by verifying the definition

Ex. 42. A ={ 1

n2 + 1, n ∈N

}.

[inf A = 0, max A = 1

]Ex. 43. A =

{(−1)n

n2 + 2, n ∈N

}.

[min A = −

13, max A =

12

]

5


Ex. 44. A ={x + 2

x − 3, x ∈ R, x > 3

}.

[inf A = 1, sup A = +∞

]Ex. 45. A =

{x + 2x − 2

, x ∈ R, x < 2}.

[inf A = −∞, sup A = 1

]Ex. 46. A =

{ nmn2 + m2 , (n, m) ∈N ×N \ {(0, 0)}

}.

[min A = 0, max A =

12

]Ex. 47. A =

{ nmn2 + m2 , (n, m) ∈N \ {0}

}.

[inf A = 0, max A =

12

]Ex. 48. A =

{n + mn −m

, n,m ∈N, n , m}.

[inf A = −∞, sup A = +∞

]Ex. 49. A =

{ nm

+mn, n,m ∈N \ {0}

}.

[inf A = 2, sup A = +∞

]Study the boundedness of the following numerical sets, expressing for any of them

sup, inf, max and min

Ex. 50. A ={3n + 1

n + 2, n ∈N \ {0}

}.

[min A =

43, sup A = 3

]Ex. 51. A =

{ 11 + 2−n , n ∈N \ {0}

}.

[min A =

23, sup A = 1

]Ex. 52. A =

{ 2nn! + 1

, n ∈N \ {0}}.

[inf A = 0, max A =

43

]Ex. 53. A =

{log n!

n!, n ∈N

}.

[min A = 0, max A = log

√2]

Ex. 54. A =

{n

sin(1 + nπ/2), n ∈N

}.

[inf A = −∞, sup A = +∞

]Ex. 55. A =

√n −√

n + 2n2 , n ∈N \ {0}

. [min A = −

2

1 +√

3, sup A = 0

]Ex. 56. A =

{∣∣∣∣∣(−1)n nn + 3

−15

∣∣∣∣∣ , n ∈N}.

[min A =

15, sup A =

65

]Ex. 57. A =

{∣∣∣∣n2 + sin(nπ2

)∣∣∣∣ , n ∈N

}.

[min A = 0, sup A = +∞

]Ex. 58. A =

{sin

((2n + 1)π

2

)21/(n+1), n ∈N

}.

[min A = −

√2, max A = 2

]Establish if the following numerical sets are bounded; find sup, inf, max and min, if they

exist

Ex. 59. A ={ 1

1 + 2n, n ∈N, n ≥ 1

}.

[inf A = 0, max A =

13

]Ex. 60. A =

{x ∈ R :

xx + 1

>12

}.

[A = (−∞,−1) ∪ (1,+∞); inf A = −∞, sup A = ∞

]

6


Ex. 61. A ={x ∈ R :

√

x2 − 2x <12

x}.

[min A = 2, sup A =

83

]Ex. 62. A = {x ∈ R :

√log(sin x) ∈ R}.

[A = {

π2

+ 2kπ, k ∈ Z}; inf A = −∞, sup A = +∞]

Ex. 63. A = {x ∈ R : 1 ≤ 32x+1 < 9}.[

min A = −12, sup A =

12

]Ex. 64. A =

{x ∈ R : 5 <

15

3x−3≤ 25

}.

[min A =

13, sup A =

23

]Ex. 65. A =

{1 −

(−1)n

n, n ∈N \ {0}

}.

[min A =

12, max A = 2

]

Ex. 66. A =

4

2n + 1, n ∈N, n even

2 −1

n + 1, n ∈N, n odd

.[

inf A = 0, max A = 4]

Ex. 67. Define an infinite set using a non-monotone sequence such that 0 and 1 will be the

inf and sup of the set respectively.

Ex. 68. Find inf e sup of the areas of the surfaces of the rectangles with perimeter equal to

4a, for a a positive real number, different from zero.

1.8 Domain of functions

Determine the domain of the following functions and study the boundedness of such sets.

Then trace a qualitative graph of the functions themselves.

Ex. 69. f (x) =√

x2 − 1.

Ex. 70. f (x) =

√1 − xx + 2

.

Ex. 71. f (x) =4

√|1 − x|x + 2

.

Ex. 72. f (x) = log1/2(1 − |x|).

Ex. 73. f (x) = 6√

log1/3(2 − |x|).

Ex. 74. f (x) =√

log2(x2 − 2x − 5) − 1.

Ex. 75. f (x) =√

log3(2x + 2) − log3 x.

Ex. 76. f (x) =

√log3(

x + 2x

).

7


Ex. 77. f (x) =√

log3(x + 1) − log9(x + 2) + 1.

Ex. 78. f (x) = 2(x+2)/(x2−3x−4).

Ex. 79. f (x) = log5(62x− |4 · 6x

− 1|).

Ex. 80. f (x) = cos(2x − 1

x + 1

).

Ex. 81. f (x) =

√cos

(2x − 1x + 1

).

Ex. 82. f (x) =(cos

(2x − 1x + 1

)−

12

)1/4.

Ex. 83. f (x) =1

sin x + cos x.

Ex. 84. f (x) = 2 log3(sin x + 2 cos x).

Ex. 85. f (x) = log3(sin x + 2 cos x)2.

Ex. 86. f (x) = log23(sin x + 2 cos x).

Ex. 87. f (x) = arccos(x + 1x − 1

).

Ex. 88. f (x) = arcsin( x + 1|x| − 1

).

Ex. 89. f (x) =(log4(sin x)

)1/2.

Ex. 90. f (x) =[2

4√

1−log7(x2+x)− (x2 + x)

]1/2.

Ex. 91. Indicated by D the domain of any function of the exercises in the paragraph 1.8,

determine the set of the interior points D̊ of D and the set of its boundary points ∂D.

Moreover, say if such sets are oper or closed and study their boundedness.

Ex. 92. Determine the set of the images (range) for any function of the exercises in the

paragraph 1.8, and the set of the accumulation points of such sets.

Ex. 93. Given two functions f , g : A ⊆ R→ R, show the following implications:

1. f , g increasing =⇒ f + g increasing;

2. f , g decreasing =⇒ f + g decreasing;

3. f increasing and g strictly increasing =⇒ f + g strictly increasing;

4. f decreasing and g strictly decreasing =⇒ f + g strictly decreasing.

8


Ex. 94. Establish under which conditions the following implication is true:

f , g increasing (or decreasing) =⇒ f · g increasing (or decreasing).

Ex. 95. Furnish an example such that the result of the exercise 94 is, in general i.e. without

further hypothesis, false.

Ex. 96. Show that if f : A ⊆ R→ R is invertible, then

f increasing (decreasing) =⇒ f−1 increasing (decreasing).

Ex. 97. Let f : A ⊆ R→ R be such that 0 < f (A) and increasing. Determine if1f

is increasing

or decreasing.

Ex. 98. Let f , g : A ⊆ R→ R two injective functions. Is the function f + g invertible?

Ex. 99. Let f : X→ Y and g : V →W and let moreover f (X)∩V , ∅. If f and g are invertible

functions, is the composition f ◦ g an invertible function?

Ex. 100. Furnish three different examples of functions f : X→ X such that f ≡ f−1.

1.9 Invertibility of functions

Study the invertibility of the following functions in their natural definition set.

Ex. 101. f (x) = 2x + x.

Ex. 102. f (x) = −x + log1/2 x.

Ex. 103. f (x) = x2 + log3(1 + x).

Ex. 104. f (x) =5x

1 + 5x + x3.

Ex. 105. f (x) = x|x| + 1.

Ex. 106. f (x) =

1

x − 1if x > 1

x + a if x ≤ 1al variare di a ∈ R.

Ex. 107. f (x) =

x2 + ax if x ≤ 0

−1x

if x > 0for any a ∈ R.

Ex. 108. f (x) =

x3 if |x| ≥ 1

ax if |x| < 1.for any a ∈ R.

9


Ex. 109. Let f : X→ Y and g : V →W be two invertible functions such that it is well defined

the composed function g ◦ f . Call f−1 and g−1 their inverses respectively, show that

(g ◦ f )−1 = f−1◦ g−1.

Verify that the following functions are invertible; then determine the inverse of any of

them, specifying its domain.

Ex. 110. f (x) = x|x| + x.

Ex. 111. f (x) = x(x − 2), x ≤ 0.

Ex. 112. f (x) = log1/2(1 − x3).

Ex. 113. f (x) =3x+1

1 + 3x+1.

Ex. 114. f (x) =√

e2x + ex + 1.

Ex. 115. f (x) = sin3(

x2

x2 + 1

), x ≤ 0.

Ex. 116. f (x) = arccos(log2 x).

Ex. 117. f (x) = tan(x3 + 1),π2< x3 + 1 <

32π.

Ex. 118. f (x) = arctan(x3 + 1).

Ex. 119. f (x) = arcsin(√

x2 + 1), x < 0.

10

2 Complex numbers

2.1 Elementary properties of complex numbers

Determine z̄, Im(z) e |z| in the following cases:

Ex. 120. z = 3 − 4i.[3 + 4i, −4, 5

]Ex. 121. z = (2 − i)(−3 + 2i).

[− 4 − 7i, 7,

√65

]Ex. 122. z = (1 − i)(3 − 7i).

[− 4 + 10i, −10, 2

√29

]Ex. 123. z = (2 − 3i)2.

[− 5 + 12i, −12, 13

]Ex. 124. z =

(1 − 2i2

)2.

[−

34

+ i, −1,54

]Ex. 125. z =

23 − i

.[3 − i

5,

15,

√25

]Ex. 126. z =

2 − 3i1 − i

.[5 + i

2, −

12,

√132

]Ex. 127. z = (1 − 2i)3.

[− 11 − 2i, 2, 5

√5]

Ex. 128. z =(1 − i)3

2 − i.

[−2 + 6i5

, −65, 2

√25

]Ex. 129. z =

(1 + 2i)4

(1 − i)2 .[12 +

72

i, −72,

252

]Compute the absolute value and argument of z in the following cases:

Ex. 130. z = 1 + i.[√

2,π4

+ 2kπ, k ∈ Z]

Ex. 131. z = 1 −√

3i.[2,−

π3

+ 2kπ, k ∈ Z]

Ex. 132. z =

√3

3+

i3.

[23,π6

+ 2kπ, k ∈ Z]

Ex. 133. z = i(1 − i).[√

2,π4

+ 2kπ, k ∈ Z]

Ex. 134. z =2

1 +√

3i.

[1, −

π3

+ 2kπ, k ∈ Z]

11


Ex. 135. z =3√

3 + i.

[32, −π6

+ 2kπ, k ∈ Z]

Ex. 136. z = (1 − i)(√

3 + i).[2√

2, −π12

+ 2kπ, k ∈ Z]

Ex. 137. z =1 − i√

3 + i.

[ 1√

2, −

512π + 2kπ, k ∈ Z

]Ex. 138. z =

√3 − i

1 +√

3i.

[1, −

π2

+ 2kπ, k ∈ Z]

Ex. 139. z = (1 − i)12.[64, π + 2kπ, k ∈ Z

]Ex. 140. z =

(1 +

i√

3

)14

.[214

37 ,π3

+ 2kπ, k ∈ Z]

Ex. 141. z =i324− i261

i145 + i492.

[1, −

π2

+ 2kπ, k ∈ Z]

Ex. 142. z =1 − i1039

i2048 − i1457.

[1,π2

+ 2kπ, k ∈ Z]

Ex. 143. z =cos 2θ − i sin 2θ

sinθ + i cosθ.

[1, −θ −

π2

+ 2kπ, k ∈ Z]

Ex. 144. z =12 sin 2θ + i cosθ

1 − i sinθ.[

| cosθ|,π2

se cosθ > 0, −π2

se cosθ < 0, not determinate if cosθ = 0]

2.2 Roots of complex numbers

Compute the following roots of the complex numbers:

Ex. 145.√

1 +√

3i.[±

√3 + i√

2

]Ex. 146.

3√

1 +√

3i.[

3√2(cos

π9

+ i sinπ9

),

3√2(cos

7π9

+ i sin7π9

),

3√2(cos

5π9− i sin

5π9

) ]Ex. 147.

√1 + i1 − i

.[±

1 + i√

2

]Ex. 148. 4

√(1 − i)3 + (1 + i)3.

[1 + i, 1 − i, −1 + i, −1 − i

]Ex. 149.

√2

(1 − i)2

(1 + i)3 .[±

4√2(cos

3π8

+ i sin3π8

)= ±

√√

2 − 1 + i√√

2 + 1√

2

]

12


Compute the following expressions containing complex numbers:

Ex. 150.

√((1 + i)2

− (1 − i)3)− 2

(3 − i)2 + 6i.

[±

1 + i2

]Ex. 151.

3

√1 +

(2 − 3i)2 + 7i5

.[i, ±√

32−

i2

]Ex. 152. 4

√(1 + 2i)(3 − i)

5−

(1 − i)(1 + 3i)2

.[1 + i√

2,

1 − i√

2,−1 + i√

2,−1 − i√

2

]Ex. 153. 3

√(2 − i)2

− 3(1 + 2i)3 + 11

.[

3√2,−1 ±

√3

3√4

]Ex. 154.

√−1 +

3√i.[0, −2i,

3i2±

√3

2, −

i2±

√3

2

]2.3 Complex equations

2.3.1 Algebraic complex equations

Determine the solutions of the following algebraic equations:

Ex. 155. z2 + 2z + 2 = 0.[z = −1 + i, −1 − i

]Ex. 156. z2

− 6z + 13 = 0.[z = 3 + 2i, 3 − 2i

]Ex. 157. 4z2

− 4z + 17 = 0.[z =

12

+ 2i,12− 2i

]Ex. 158. z3 + 3z2 + z − 5 = 0.

[z = 1, −2 + i, −2 − i

]Ex. 159. z4 + 4 = 0.

[z = 1 + i, 1 − i, −1 + i, −1 − i

]Ex. 160. z2

− iz + 6 = 0.[z = −2i, 3i

]Ex. 161. 4z2

− 2(1 − i)z − i = 0.[z =

12, −

i2

]Ex. 162. 2z2 + iz + 3 = 0.

[z = i, −

3i2

]Ex. 163. 2z2

− 5iz − 2 = 0.[z = 2i,

i2

]Ex. 164. 6z2

− (3 + 2i)z + i = 0.[z =

12,

i3

]Ex. 165. 8z2

− 2(16 + i)z + 5(5 + 2i) = 0.[z = 1 +

i2, 3 −

i4

]Ex. 166. 2z3

− (2 − i)z2 + (1 − i)z − 1 = 0.[z = 1,

i2, −i

]Ex. 167. 2z3 + (2 + 5i)z2 + (3 + 5i)z + 3 = 0.

[z = −1, −3i,

i2

]

13


2.3.2 Non-algebraic complex equations

Ex. 168. z|z3| + 16 = 0.

[z = −2

]Ex. 169. z2

|z2| + 16 = 0.

[z = 2i, z = −2i

]Ex. 170. z3

|z| + 16 = 0.[z = −2, z = 1 +

√3i, z = 1 −

√3i]

Ex. 171. z2(1 + |z2|) = −20.

[z = 2i, z = −2i

]Ex. 172. z2(1 − |z2

|) = −20.[z =√

5, z = −√

5]

Ex. 173. z2(4 − |z2|) = 5.

[z =√

5i, z = −√

5i]

Ex. 174. z2(4 − |z2|) = 4.

[z =√

2i, z = −√

2i√

2 + 2√

2i, −√

2 + 2√

2i]

Ex. 175. z2(4 − |z2|) = 3.

[z = 1, z = −1, z =

√3, z = −

√3, z =

√2 +√

7i, −√

2 +√

7i]

Ex. 176.z2

1 + |z2|= −

12.

[z = i, z = −i

]Ex. 177.

z2

1 + |z2|= −2.

[Nessuna soluzione

]Ex. 178.

z2

1 − |z2|= −

12.

[z =

√23

i, z = −

√23

i]

Ex. 179.z2

1 − |z2|= −2.

[z =

i√

3, z = −

i√

3

]Ex. 180.

z4

|z2|== −8.

[z = 2 + 2i, z = 2 − 2i, z = −2 + 2i, z = −2 − 2i

]Ex. 181.

z2

|z4|== −

18.

[z = 2

√2i, z = −2

√2i]

Ex. 182.z4

|z6|== 8.

[z =

1

2√

2, z = −

1

2√

2, z =

i

2√

2, z = −

i

2√

2

]Ex. 183.

z2− |z2|

4 + |z2|+ 1 = 0.

[No solution

]Ex. 184.

z2− |z2|

4 + |z2|+

12

= 0.[z =√

2i, z = −√

2i]

Ex. 185.z2− |z2|

4 + |z2|+

14

= 0.[z = (1 +

√3)i, z = (1 −

√3)i, z = −(1 +

√3)i, z = −(1 −

√3)i

]Ex. 186.

z2− |z2|

4 + |z2|= 0.

[Im(z) = 0

]Ex. 187. z(2 + |z2

|) =3z̄.

[|z| = 1

]

14

3 Limits of one real variable funtions

3.1 Check, using the definition, the following limits

Verify the definition of limit in the following cases:

Ex. 188. limx→1

x = 1.

Ex. 189. limx→+∞

1x

= 0.

Ex. 190. limx→3

(2x + 1) = 7.

Ex. 191. limx→2

x2 = 4.

Ex. 192. limx→0

1x2 = +∞.

Ex. 193. limx→0

1x3 @.

Ex. 194. limx→1

3x = 3.

Ex. 195. limx→π/2

sin x = 1.

Ex. 196. limx→(π/2)−

tan x = +∞.

Ex. 197. limx→1+

x − [x] = 0.

Ex. 198. limx→1−

x − [x] = 1.

Ex. 199. limx→0+

log1/2 x = +∞.

Ex. 200. limx→+∞

x + 22x + 2

=12.

Ex. 201. limx→+∞

sin1x

= 0.

3.2 Computation of limits

Calculate, if they exist real or infinite,the following limits:

Ex. 202. limx→2

(x2 +

1x

).

[92

]

15


Ex. 203. limx→+∞

x + x2

x3 + 1.

[0]

Ex. 204. limx→0

x sin x1 − cos x

.[2]

Ex. 205. limx→+∞

(√x2 + 4 − x

).

[0]

Ex. 206. limx→+∞

(√2x + x2 − x

).

[1]

Ex. 207. limx→+∞

log2(x + x2)log3 x − 1

.[2 log2 3

]Ex. 208. lim

x→0

sin x − xx9/10

.[0]

Ex. 209. limx→4

√

x2 + 1 −√

17x − 4

.[ 4√

17

]Ex. 210. lim

x→0+41/x.

[+∞

]Ex. 211. lim

x→0−41/x.

[0]

Ex. 212. limx→0

sin x −√

x1 − cos 4√x

.[− 2

]Ex. 213. lim

x→0

sin x − x2√

1 − cos x2.

[+∞

]Ex. 214. lim

x→π/22(sin x−1)/x4

.[1]

Ex. 215. limx→0

sin2 x2

+ cos x − 1

x2 .[0]

Ex. 216. limx→+∞

log4

(x + 1x − 1

).

[0]

Ex. 217. limx→0

log3(x + 1)x

.[

log3 e]

Ex. 218. limx→0

log1/2 cos x

x2 .[

log4 e]

Ex. 219. limx→1+

(sin x)1/ log2 x.[0]

Ex. 220. limx→+∞

x3

2x .[0]

Ex. 221. limx→+∞

log3 xx

.[0]

16


Ex. 222. limx→+∞

x3

2log3(log2 x).

[+∞

]Determine domain and image of the following functions, indicating if they are periodic

and even or odd.

Ex. 223. f (x) =√

2 sin2 x + cos x − 1.

Ex. 224. f (x) = log3(sin3 x − cos3 x).

Ex. 225. f (x) = log1/2(| sin 2x| + cos x).

Ex. 226. f (x) = 4(sin x+cos x)/(sin x−cos x).

Ex. 227. f (x) =1

2sin x − 3cos x .

Ex. 228. f (x) = |x|α sin1x3 , al variare di α ∈ R.

Ex. 229. f (x) = arcsin( 2 + ex

e2x − 3

).

Ex. 230. f (x) =4√

tan2(x2 + 1) − tan(x2 + 1) − 6.

Ex. 231. f (x) =5x + 5−x

2.

Ex. 232. f (x) = arctan5x− 5−x

2.

Draw a qualitative graph of the functions studied in the exercises 223, 226, 228, 231 and

232 above.


Ex. 233. limx→+∞

(x + 5)

√x + 1x − 1

− x.[6]

Ex. 234. limx→+∞

x(log(x + 1) − log x).[1]

Ex. 235. limx→+∞

(x3− 2x + 1

x2 + x3

)(2x2+1)/(x−3)

.[e−2

]Ex. 236. lim

x→0

log cos xsin 2x2 .

[−

14

]Ex. 237. lim

x→0+(sin x)x2+3x log x.

[1]

Ex. 238. limx→0

log(1 + sin x)sin 2x + x2 log x

.[12

]

17


Ex. 239. limx→0

e2x−3− e−3

sin x.

[2e−3

]Ex. 240. lim

x→0

sin(√

1 + x2 − 1)

x.

[0]

Ex. 241. limx→0

(log(1 + x) + sin x + x

x + x2

)2

.[9]

Ex. 242. limx→0

e−1/x2+ log

(1 + x1/5

− sin 3√x)

3√x − 2 5√x.

[−

12

]Ex. 243. lim

x→+∞

sin(x5/3x)x42−x .

[0]


Ex. 244. limx→1

√x − cos(x − 1)

log x.

[12

]Ex. 245. lim

x→2

(sin

πx4

)1/ log(3−x).

[1]

Ex. 246. limx→1|x − 1|x−1.

[1]

Ex. 247. limx→0+

x1/ log x.[e]

Ex. 248. limx→+∞

(cos(1/x)cos(2/x)

)(x2+1)/x

.[1]

Ex. 249. limx→0+

(2xx− 1)1/

√x− 1

√x log x

.[2]

Ex. 250. limx→+∞

x2((e1/x + 1)1/2− cos(1/x)).

[+∞

]Ex. 251. lim

x→+∞x2((2e1/x2

− 1)1/2− cos(1/x)).

[32

]Ex. 252. lim

x→3

e−1/(3−x)2+ e(4 − 3 cos(x − 3))1/5

− e√

4−x√1 − cos(x − 3)

.[−

e√

2

]Ex. 253. lim

x→+∞sin(1/x) · log(x2 + e1/x + 2x2/(x+1)).

[log 2

]

Ex. 254. limx→+∞

1

log10(x2 + x + 1)

sin1

xx + 1

log10(x3 + x + 1)

−1

.[(32

)10]

Ex. 255. limx→0

arcsin√

x√cos 4√x − 1

.[− 4

]

18


Ex. 256. limx→0

(1 + sin x)1/ arctan x.[e]


Ex. 257. limx→+∞

x2 + sin x

x + log(x + e2x2

) . [12

]Ex. 258. lim

x→+∞

(x4e−x + sin(1/x2) + 1

)√1+2x4

.[e√

2]

Ex. 259. limx→+∞

(√x + x3 − x

)log

( √4x + 1

2√

x + 3

)x arctan x

.[−

3π

]Ex. 260. lim

x→+∞

xarctan x− xπ/2

(1 + x)π/2+1/√

log x.

[0]

Ex. 261. limx→+∞

xarctan x− xπ/2

(1 + x)π/2−1.

[−∞

]

Ex. 262. limx→0+

e−1/x + x2 +1

log2 x+ x log

(e−1/x + e−2/x

)+ 1

ex − 1.

[+∞

]Ex. 263. lim

x→0+

x sin x − cos x + ex2/2√

1 − cos x · arcsin x.

[ 4√

2

]Ex. 264. lim

x→1

(x2− 2x + 1) tan(x − 1) − sin3(x − 1)√

cos(x − 1) − 1.

[0]

Ex. 265. limx→0+

x(cos√

x3 − 1)

+ sin2 x3/4

x3e−1/√

x +√

x(ex2− 1

) .[

+∞]

Ex. 266. limx→0+

x(cos√

x3 − 1)

+ sin2 x3/4

x3e−1/√

x +(ex2− 1

)/√

x.

[1]

Ex. 267. limx→0+

log | log x| + log x

log(1 + xlog x

) .[0]

Ex. 268. limx→1

e3x−x2− e2 cos(x − 1) − x + 1log (sinπx/2)

.[(2eπ

)2]


Ex. 269. limx→0+

xx; limx→0+

xxx; lim

x→0+xxxx

.[1; 0; 1

]

Ex. 270. limx→0+

n times︷︸︸︷xx.

..x

.[1 if n is even, 0 if n is odd

]

19


Ex. 271. limx→1

((x − 1)2

sin(πx)(e − ex))log x.

[1]

Ex. 272. limx→+∞

cos(1/x) − e−1/x2

(√x4 − x2 − x2

)log

√x2 + 2x2 + 1

.[− 2

]

Ex. 273. limx→0+

sin(ex2− cos x + 2 sin x2

√1 + 2 sin2 x)

2 sin2 x.

[74

]

Ex. 274. limx→0+

√ex2− cos x + 2 sin x3

√

1 + 2 sin x3

2 sin3 x.

[+∞

]Ex. 275. lim

x→0+

√1 + x sin x −

√cos 2x

tan2(x/2).

[6]

Ex. 276. limx→0

log(2 − cos x)(2 − cos x)1/x2sin x2

sin2 x2.

[ √e2

]Ex. 277. lim

x→0

log(cos2 x)(x −√

x2 + 3x + 1)1 + e−1/x − cos x

.[2]

Ex. 278. limx→0

((sin x + 2)2 log(sin x + 1)

)(√

1+3x2−1)/x2

.[0]

Ex. 279. limx→+∞

log(

e−1/x

x4+ 1

)+ sin3(1/x)

log(

2 + x3

x3

) .[12

]Arrange in growing order of infinity (infinitesimal) the following functions and sequences,

after having determined the order of infinite (infinitesimal), if it exits as a real number.

Ex. 280. For x→ +∞: a)ex

x2 , b) x log x, c)x2

log x, d)

1sin(1/x)

.[d, b, c, a. ord d=1

]Ex. 281. For n→ +∞: a) 2n, b) n!, c) nn, d)

(32

)n2

.[a, d, b, c

]Ex. 282. For x→ +∞: a) xx, b) x log2 x, c) x2 log x, d)

x5 + x3 + 2x2 + 1

logx + 1

x.[

b, d, c, a. ord d=2]

Ex. 283. For x→ 0+: a)1

log x, b) x2, c)

3√1 − cos x√

arcsin x, d) log x · arcsin x.[

a, c, d, b. ord b=2, c=16

]Ex. 284. For x→ 0+: a) log x, b) log | log x|, c)

1x log x

, d)1

log(1 + x).[b, a, c, d. ord d=1

]

20


Ex. 285. For x→ 1+: a) e−1/(x−1)2, b) 10√x − cos(x − 1), c) sin3 3√

x2 − x, d)x − 1

log20(x − 1).[

c, d, b, a. ord b=1, c=13

]Ex. 286. For x→ 2+: a)

1(x − 2)3/2

, b)1

(x − 2)3/4 log(x − 2), c) e

√x−2/ sin(x−2),

d) (x − 2)1/(2−x).[a, b, c, d. ord a= 3

2

]Ex. 287. For x→ 0+: a) x arctan x, b)

1 − cos xlog x

, c) xx− 1, d) sin3 4√x.[

d, c, b, a. ord a=2, d=34

]Arrange in growing order of infinity (infinitesimal) the following functions and sequences,

after having determined the order of infinite (infinitesimal), if it exits as a real number.

Ex. 288. For x→ +∞: a) x2, b) log(1 + x3 + ex3), c)

x2

x + 1, d)

(x2

x + 1

)1+1/√

log x

.[c, d, a, b. ord a=2, b=3, c=1

]Ex. 289. For n→ +∞: a)

√n

n2 + 1, b)

1n log n

, c)log2 n

n, d)

n!(n + 1)! − (n − 1)!

.[c, d, b, a. ord a= 3

2 , d=1]

Ex. 290. For n→ +∞: a) ( n√n− 1)−1, b) n(√

3 + n2 −n), c) (cos(1/n)− 1) · 2n3/(n+1), d) nn.[a, b, c, d. ord b=2

]Ex. 291. For x→ 0+: a)

x2(1 − cos x)2

log(1 + sin4 x), b) log(x+1), c) x log x, d) sin(x log(1+x)) ·log x.[

c, b, d, a. ord a=2, b=1]

Ex. 292. For x→ +∞: a)x2 log(2 − cos(1/x))

sin2(1/x), b)

x√

x

x100, c) x2 log

(x2 + 1

x

),

d) x log100(1 + x).[d, a, c, b. ord a=2

]Ex. 293. For x→ 3+: a) (e

(x−3)2

(3−x)(x+1)3 − 1) sin(x − 3)9/4, b) sin3(x − 3), c) (x − 3)3 log(x − 2),

d) (x − 3)3 log10(x − 3).[d, b, a, c. ord a= 13

4 , b=3, c=4]

Ex. 294. For x→ 0+: a) x log(1 + x2), b) x2−x/(x2+1), c)

3√

x2 + x4√

x2 + 2x

25

, d) x3 log10 x.[b, c, d, a. ord a=3, b=2, c=4, c= 25

12

]Ex. 295. For x→ 0+: a) x arctan

√x, b)

(1 − cos x)2√

x + 1√

x4 + 1 log(1 + x2), c) x2 log

(x2 + 1

x

)e√

x,

d) sin(x3 log x).[a, c, b, d. ord a=3

2 , b=2]

21


Calculate the limit of the following sequences:

Ex. 296. limn→+∞

en2

nn .[

+∞]

Ex. 297. limn→+∞

en3/2

nn2 + en.

[0]

Ex. 298. limn→+∞

√

n2 + n3 − n + sin n4√

1 + n5 + 2n6.

[ 4√22

]Ex. 299. lim

n→+∞

2(1+log1/2 n)

n1/2.

[0]

Ex. 300. limn→+∞

(log(n2 + 1) − log n − log(n + 1))√

1 + n2.[− 1

]Ex. 301. lim

n→+∞

2n− 3n

4n .[0]

Calculate the limit of the following sequences:

Ex. 302. limn→+∞

n√

(n2 + 1) sin(1/n).[1]

Ex. 303. limn→+∞

nn

(n!)!.

[0]

Ex. 304. limn→+∞

n√

n!.[

+∞]

Ex. 305. limn→+∞

∣∣∣∣logn

n + 1

∣∣∣∣ 1−2√

n+1n+√

n .[1]

Ex. 306. limn→+∞

(1 +

n!nn

) (n−1)nn(n+1)!

.[

e√e]

Ex. 307. limn→+∞

√

n2 + 1 arcsin(e−n +

1n2 + n

).

[0]

Ex. 308. limn→+∞

(1 + cos(1/n) − cos(2/n))−(arcsin(1/n))n2

.[1]

Ex. 309. limn→+∞

(n2 + n3 + 3√

n + n + n3 − 1e−1/n)n.

[1]

Ex. 310. limn→+∞

n6 + en log n + 2n4 arcsin(1/n)

nn − n! + en3 .[1]

Ex. 311. limn→+∞

n2√n3 + 1 + en2 .

[e]

Ex. 312. limn→+∞

n√

en + sin(πn/2).[e]

Ex. 313. limn→+∞

n√

2 + sin n.[@]

22


*Ex. 314. Let {an} be a positive terms sequence such that

limn→+∞

logan

an+1≥ 0.

Give at least two counterexamples showing that from this relation is not possible deduce that

limn→+∞

an = +∞

Moreover, say under which further hypothesis the result would be true.

Ex. 315. Using the comparison theorem, show that

limn→+∞

1n2 + 1

+1

n2 + 2+ · · · +

1n2 + n

= 0.

*Ex. 316. Let {an} be a positive terms sequence. Show that

limn→+∞

an+1

an= r ≥ 0 ⇒ lim

n→+∞

n√an = r

Use the sequence an = e1/n + sin(πn/2) + 1 to show that in general the converse is not true.

*Ex. 317. Show, exhibiting a counterexample, that if {an} is a non-negative terms sequence,

then

limn→+∞

a1/nn = l ; lim

n→+∞

an

ln= 1

Moreover, show that if limn→∞

a1/nn = l > 1 then an → +∞ for n→ +∞.

23

4 Study of functions of one real variable

4.1 Asymptotes

Determine the possible asymptotes (vertical, horizontal, oblique) for the following functions,

after having indicated their domain. Moreover, calculate the limit of the functions to the

boundary points of their domain.

Ex. 318. f (x) =x + 1

3 − 2x.

Ex. 319. f (x) =1

x(x − 2).

Ex. 320. f (x) =

√

x4 + 1x − 2

.

Ex. 321. f (x) = x log(1 + x).

Ex. 322. f (x) =x

x2 + 1.

Ex. 323. f (x) =x

x2 − 1.

Ex. 324. f (x) = x arcsin1

x + 1.

Ex. 325. f (x) = elog2(x/(x−1))+log(3x−3)+2.

Ex. 326. f (x) = log(1 − 3ex + 2e2x).

Ex. 327. f (x) = x ex/(x2−1).

Ex. 328. f (x) = x√

cosx

x2 + 1.

Ex. 329. f (x) =log |x|

3 + log |x|+√

x2 + 2x.

Ex. 330. f (x) = x arctan x. (Use the formula arctan x + arctan1x

=π2, x > 0).

Ex. 331. f (x) = x1+1/ log x.

Ex. 332. f (x) = x1+log x/√

1+log2 x.

Ex. 333. f (x) =x2

x4 − 1e−1/x2

.

24


4.2 Continuity and derivability

Determine the domain and the set of continuity of the following functions.

Ex. 334.

f (x) =

x − [x] − 1, x ≤ 2

x − [x], x > 2.

Ex. 335. f (x) = [x] +√

x − [x].

Ex. 336. f (x) = 41/ sin x.

Ex. 337. f (x) =sin(log x)

log x.

Ex. 338.

f (x) =

sin(cot x), x , kπ, k ∈ Z

0, x = kπ, k ∈ Z.

Ex. 339. Determine a ∈ R such that the following function result to be continuous

f (x) =

x2− 1

x + 1, x , −1

a, x = −1.

Ex. 340. Say if it is possible to apply the Weierstass theorem about the existence of the

extremes to the following function

f (x) =

x, 0 ≤ x < 1

1 − x, 1 ≤ x ≤ 3.

Determine the set of continuity and the set of derivability of the following functions and

calculate their derivative.

Ex. 341. f (x) = tan 2x.

Ex. 342. f (x) = e2x− e−2x.

Ex. 343. f (x) = 32x.

Ex. 344. f (x) = xx2+1.

Ex. 345. f (x) =2x + 3x − 4

.

Ex. 346. f (x) =

√2x + 3x − 4

.

Ex. 347. f (x) =

√x

x2 + 1.

25


Ex. 348. f (x) = (arcsin x)3.

Ex. 349. f (x) = esin x.

Ex. 350. f (x) = arctan( x1 − x2

).

Ex. 351. f (x) = log tan x.

Ex. 352. f (x) = arcsin(

11 +√

x

).

Ex. 353. f (x) = arcsin(

x2

x2 − 1

).

Ex. 354. f (x) = x e1/(1−x).

Ex. 355. f (x) = 2arccos 3x.

Ex. 356. f (x) = log 2|x|.

Ex. 357. f (x) =log x

3 − 2 log(2x).

Ex. 358. f (x) = |x|x + ex.

Ex. 359. f (x) =√

x2 + x4 arctan x.

Ex. 360. f (x) =√

1 − cos x.

Ex. 361. f (x) =

√log

(x2

x2 − 1

).

The same work (determination of continuity, derivability and calculation of the derivative)

is recommended also for the functions in the exercises in paragraphs 1.8 and 4.1.

4.3 Invertibility and derivative of the inverse function

Verify the invertibility of the following functions and determine the domain of derivability

of the respective inverse functions.

Ex. 362. f (x) = 2x + log x.

Ex. 363. f (x) = −x + e−2x.

Ex. 364. f (x) = x|x| + log(1 + x).

Ex. 365. f (x) = x + sin x.

26


Ex. 366. f (x) = x√|x| + arctan x.

Ex. 367. f (x) =5√1 − x − cos x.

Ex. 368. For any of the following function f (x), determine:

f ′(2), f ′(1), f ′(1 + log 2), f ′(π2

+ 1), f ′(1 +π4

), f ′(0).

Moreover, write the equation of the tangent line passing for the point indicated.

Ex. 369. Use the mean value theorem to show that

| sin x − sin y| ≤ |x − y|, x, y ∈ R .

4.4 Critical points

Determine the possible critical points for the following functions.

Ex. 370. f (x) =x

x2 + 1.

Ex. 371. f (x) =x

x2 − 1.

Ex. 372. f (x) =log x

x.

Ex. 373. f (x) = xe−1/x.

Ex. 374. f (x) =√

x∣∣∣∣∣1 +

1log x

∣∣∣∣∣ .Ex. 375. f (x) = x log x.

Ex. 376. f (x) = x3 + x2− x.

Ex. 377. f (x) =√−x(x + 1).

Ex. 378. f (x) = ex(32|x| +

12

(3x − 8)).

Ex. 379. f (x) = ((2 − x)6)log |x−2|.

4.5 Derivability and Monotony

Determine the intervals of monotony for the functions in the paragraph 4.4.

27


4.6 Taylor and Mac Laurin Polynomials

Determine the Mac Laurin polynomial of the following functions to the indicated order.

Ex. 380. f (x) = sin(x2), to the order 4.

Ex. 381. f (x) =√

1 + 2x, to the order 3.

Ex. 382. f (x) = log(1 + x3), to the order 8.

Ex. 383. f (x) = sin2 x, to the order 4.

Ex. 384. f (x) = ex+1, to the order 5.

Determine the Taylor polynomial, centered in x0 and to the indicated order, for the

following functions.

Ex. 385. f (x) = ex, x0 = 2, to the order 3.

Ex. 386. f (x) = cos x, x0 = 3, to the order 4.

Ex. 387. f (x) = log(1 + x), x0 = 2, to the order 3.

Ex. 388. Determine the Mac Laurin polynomial of order 4, for the function

f (x) = log(1 + x sin x) .

Determine the Mac Laurin polynomial of order 5, for the following functions.

Ex. 389. f (x) = (1 + x)ex.

Ex. 390. f (x) = x sin x + cos x.

Ex. 391. f (x) = sin x · log(1 + x).

4.7 Using Taylor polynomials for the calculation of limits

Calculate the following limits.

Ex. 392. limx→+∞

x3

x + 1

(e1/(x+1)

− 1)− x.

[−

32

]Ex. 393. lim

x→0+

12

x sin x + cos x − ex4

x2 log(1 + x2).

[−

2524

]Ex. 394. lim

x→1+

e−1/(x+1) +√

xx − x log x(x log (x cos(x − 1))

)2 .[

+∞]

Ex. 395. limx→+∞

x5 + x2 log x

x3 + x6 log(2 arctan x

π

)−

2π

x5

[−π4

]

28


4.8 Uniform Continuity

Ex. 396. Verify, using the definition, that f (x) = x2 is not an uniform continuous function

over X = [1, +∞).

Ex. 397. Establish if f (x) =arctan x

xis a uniform continuous function over the following

domains:

D1 = (0, +∞); D2 = (1, +∞); D3 = [1, +∞); D4 = (−∞, −1) ∪ (2, +∞).

Ex. 398. Verify if f (x) = x − log x results to be a Lipschitz function over the domain

D = [1, +∞).

Verify if the following functions result to be uniformly continuous over their domain:

Ex. 399.

f (x) =

xe−1/|x|, if x , 0,

0, if x = 0.

Ex. 400.

f (x) =

2 sin x + 1, if x < 0,

log(e(2x + 1)), if x ≥ 0.

Ex. 401. f (x) = sin(esin x)

For any of the following functions determine a ∈ R such that they result to be continuous.

Then check if, for such an a, the functions result to be also uniformly continuous throughout

their domain of definition.

Ex. 402.

f (x) =

a(ex− 1), if x < 1,

e−x, if x ≥ 1.

Ex. 403.

f (x) =

log x

x+ e1−x, if x > 1,

a, if x = 1,√

2 − x +π4− arctan x, if x < 1.

Ex. 404.

f (x) =

√

x2 − 2x + 2, if x ≤ 0,a log(x + 1)

x, if x > 0.

29


Ex. 405.

f (x) =

2 sin x2−

1√log(1 + x) + 1

, if x > 0,

a, if x = 0,

x(ex + 1) − 1, if x < 0.

30

5 Integrals of one-variable functions and numerical series

5.1 Immediate indefinite integrals (primitives)

Calculate the following indefinite integrals (primitives).

Ex. 406.∫

14√

x3dx.

[4x1/4 + c

]Ex. 407.

∫ √3qx dx, q ∈ R+.

[23

(3q)1/2x3/2 + c]

Ex. 408.∫

(a2/3− x2/3)3 dx, a ∈ R.

[a2−

95

a4/3x5/3 +97

a2/3x7/3−

13

x3 + c]

Ex. 409.∫

Pn(x) dx, Pn(x) =

n∑k=0

akxk, ak ∈ R.[ n∑

k=0

ak

k + 1xk+1 + c

]

Ex. 410.∫ n∑

k=0

αkeβkx dx, αk, βk ∈ R, βk , 0.[ n∑

k=0

αk

βkeβkx + c

]

Ex. 411.∫ n∑

k=0

αk sin βkx dx, αk, βk ∈ R, βk , 0.[−

n∑k=0

αk

βkcos βkx + c

]Ex. 412.

∫x2− 3x + 1

xdx.

[12

x2− 3x + log |x| + c

]Ex. 413.

∫3 +√

x5√

x2dx.

[5

5√

x3 +1011

10√

x11 + c]

Ex. 414.∫

a +√

1 − x2√

1 − x2dx, a ∈ R.

[a arcsin x + x + c

]Ex. 415.

∫x2

1 + x2 dx.[x − arctan x + c

]Ex. 416.

∫tan2 x dx.

[tan x − x + c

]Ex. 417.

∫cot2 x dx.

[− cot x − x + c

]Ex. 418.

∫1 + 2x2

x2(1 + x2)dx.

[−

1x

+ arctan x + c]

31


Ex. 419.∫

sin 2xcos x

dx.[− 2 cos x + c

]Ex. 420.

∫x5 + 1x + 1

dx.[x5

5−

x4

4+

x3

3−

x2

2+ x + c

]Ex. 421.

∫xn− an

x − adx, a ∈ R.

[xn

n+ a

xn−1

n − 1+ a2 xn−2

n − 2+ · · · + an−1x + c

]Ex. 422.

∫dx

sin2 x cos2 x.

[tan x − cot x + c

]Ex. 423.

∫cos 2x

sin x + cos xdx.

[cos x + sin x + c

]Ex. 424.

∫sin2 x

2dx.

[12

(x − sin x) + c]

Ex. 425.∫

cos2 x3

dx.[12

x +34

sin2x3

+ c]

Ex. 426.∫

1

sin2 x2

cos2 x2

dx.[2 tan

x2− 2 cot

x2

+ c]

5.2 Indefinite integrals by substitution

Calculate the following indefinite integrals using, for instance, the method of substitution of

variable.

Ex. 427.∫√

sin x cos x dx.[23

sin3/2 x + c]

Ex. 428.∫

x1 − x2 dx.

[−

12

log∣∣∣1 − x2

∣∣∣ + c]

Ex. 429.∫

1a2 + x2 dx, a ∈ R+.

[1a

arctanxa

+ c]

Ex. 430.∫

1a2 − x2 dx, a ∈ R+.

[−

12a

log∣∣∣∣a − xa + x

∣∣∣∣ + c]

Ex. 431.∫ √

a − x√

a + xdx, a ∈ R+.

[a arcsin

xa

+12

√

a2 − x2 + c]

Ex. 432.∫

1 + e−x

1 + xe−x dx.[

log |x + ex| + c

]Ex. 433.

∫1

√

a − bx2dx, a, b ∈ R+

[ 1√

barcsin

√ba

x + c]

Ex. 434.∫

x√

1 − x2dx.

[−

√

1 − x2 + c]

32


Ex. 435.∫

1

x√

5x − 7dx.

[ 2√

7arctan

√5x + 7

7+ c

]Ex. 436.

∫sinα x cos x dx, α , −1.

[ 1α + 1

sinα+1 x + c]

Ex. 437.∫

1e−x + ex dx.

[arctan ex + c

]Ex. 438.

∫cos(log x)

xdx.

[sin(log x) + c

]Ex. 439.

∫ √x

1 + xdx.

[2(√

x − arctan√

x) + c]

Ex. 440.∫

x2

(x − 1)3 dx.[

log |x − 1| −2

x − 1−

12(x − 1)2 + c

]Ex. 441.

∫x

√

a4 − x4dx, a , 0.

[12

arcsinx2

a2 + c]

Ex. 442.∫

cot xsinα x

dx, α ∈ R+.[−

1α sinα x

+ c]

Ex. 443.∫

1

x√

x2 − a2dx, a , 0.

[−

1a

arctanax

+ c]

Ex. 444.∫ √

x2 − a2

xdx, a ∈ R.

[√x2 − a2 − a arccos

ax

+ c]

Ex. 445.∫

ax + bcx + d

dx, a, b, c, d ∈ R, c , 0.[ 1c2

(a(cx + d) + (bc − ad) log |cx + d|

)+ cons.

]Ex. 446.

∫e2x√

ex − 1dx.

[23

√(ex − 1)3 + 2

√

ex + 1 + c]

Ex. 447.∫

tan xlog(cos x)

dx.[− log

∣∣∣log(cos x)∣∣∣ + c

]Ex. 448.

∫1

(a + x)(a2 − x2)1/2dx, a , 0.

[−

1a

√a − xa + x

+ c]

Ex. 449.∫

1sin x cos x

dx.[

log | tan x| + c]

Ex. 450.∫

1sin x

dx.[

log |sin x

1 + cos x| + c

]Ex. 451.

∫1

cos xdx.

[log |

1 + sin xcos x

| + c]

Ex. 452.∫

1√

x(a + x)dx, a ∈ R+

[ 2√

aarctan

√xa

+ c]

33


Ex. 453.∫

1√

a2 + x2dx, a ∈ R.

[log

∣∣∣∣√a2 + x2 + x∣∣∣∣ + c

]Ex. 454.

∫1

(a2 + x2)3/2dx, a , 0.

[ 1a2

x√

a2 + x2+ c

]Ex. 455.

∫1

(a2 + x2)5/2dx, a , 0.

[ 1a4

(x

√

a2 + x2−

13

x3√(a2 + x2)3

) + c]

Ex. 456.∫

1(a2 + x2)7/2

dx, a , 0.[ 1a6 (

x√

a2 + x2−

23

x3√(a2 + x2)3

+15

x5√(a2 + x2)5

) + c]

*Ex. 457.∫

1(a2 + x2)(2n+1)/2

dx, a , 0,n ∈N.[use the results of the previous exercises

]5.3 Indefinite integrals by parts

Calculate the following indefinite integrals using, for instance, the method of integration by

parts.

Ex. 458.∫

log xx3 dx.

[−

12x2

(log x +

12

)+ c

]Ex. 459.

∫sin3 x dx.

[−

13

(sin2 x cos x + 2 cos x

)+ c

]Ex. 460.

∫sin4 x dx.

[38

x −14

sin2 x +1

32sin 4x + c

]Ex. 461.

∫sin5 x dx.

[− cos x +

23

cos3 x −15

cos5 x + c]

Ex. 462.∫

sin xex dx.

[−

12(sin xe−x + cos xe−x) + c

]Ex. 463.

∫x3 arctan x dx.

[x4

4arctan x +

12

(x −

x3

3− arctan x

)+ c

]Ex. 464.

[ logα+1 xα + 1

+ c if α , 0, 1; log |x| + c, if α = 0; log | log x| + c if α = −1]

Ex. 465.∫

xex dx.[xex− ex + c

]Ex. 466.

∫x2ex dx.

[ex(x2

− 2x + 2) + c]

Ex. 467.∫

xnex dx, n ∈N.[ex(xn

− nxn−1 + n(n − 1)xn−2− · · · + (−1)nn!) + c

]Ex. 468.

∫x sin x dx.

[− x cos x + sin x + c

]

34


Ex. 469.∫

x cos x dx.[x sin x + cos x + c

]Ex. 470.

∫x2 sin x dx.

[− x2 cos x + 2x sin x + 2 cos x + c

]Ex. 471.

∫x2 cos x dx.

[− x2 cos x + 2x sin x + 2 cos x + c

]Ex. 472. In =

∫xn sin x dx, n ∈N,n > 1.

[Let I1 =

∫x cos x dx, In = −xn cos x + nIn−1 =

= −xn cos x + n(−xn−1 cos x + (n − 1)(−xn−2 cos x + . . . + 2∫

x cos x dx) . . .)]

Ex. 473. In =

∫xn cos x dx, n ∈N,n > 1.

[Let I1 =

∫x sin x dx, In = xn sin x − nIn−1 =

= xn sin x − n(xn−1 sin x − (n − 1)(xn−2 sin x − . . . − 2∫

x sin x dx) . . .)]

Ex. 474.∫

x sin2 x dx.[12

(−x sin x cos x +

x2

2+

sin2 x2

)+ c

]Ex. 475.

∫√

1 − x2 dx.[12

(x√

1 − x2 + arcsin x)

+ c]

Ex. 476.∫

x arcsin x dx.[x2

2arcsin x +

14

(x√

1 − x2 − arcsin x)

+ c]

Ex. 477.∫

xcos2 x

dx.[x tan x + log | cos x| + c

]Ex. 478.

∫arcsin2 x dx.

[x arcsin2 x + 2

√

1 − x2 arcsin x − 2x + c]

Ex. 479.∫

earcsin x dx.[12

earcsin x(x +√

1 − x2)

+ c]

Ex. 480.∫

sin px cos qx dx, p, q ∈ R, p , q.[ qq2 − p2 sin px sin qx +

pq2 − p2 cos px cos qx + c

]Ex. 481.

∫√

x2 + a dx, a ∈ R.[12

(x√

x2 + a + a log∣∣∣∣√x2 + a + x

∣∣∣∣ + c]

Ex. 482.∫

ex sin x dx.[12

ex(sin x − cos x) + c]

Ex. 483.∫

ex cos x dx.[12

ex(sin x + cos x) + c]

Ex. 484.∫

eαx sin x dx, α ∈ R.[ 1α2 + 1

eαx(α sin x − cos x) + c]

Ex. 485.∫

eαx cos x dx, α ∈ R.[ 1α2 + 1

eαx(sin x + α cos x) + c]

35


Ex. 486.∫

eαx sin βx dx, (α, β) ∈ R2, (α, β) , (0, 0).[ 1α2 + β2 eαx(α sin βx − β cos βx) + c

]Ex. 487.

∫eαx cos βx dx, (α, β) ∈ R2, (α, β) , (0, 0).

[ 1α2 + β2 eαx(β sin βx + α cos βx) + c

]*Ex. 488.

∫ex cosn x dx, n ∈N.

(Use the formula:

cosn x =

1

2n−1

bn/2c∑k=0

(nk

)cos(n − 2k)x, n odd

12n

(n

n/2

)+

12n−1

bn/2c−1∑k=0

(nk

)cos(n − 2k)x, n even

and the result of the exercise 487.)

*Ex. 489.∫

ex sinn x dx, n ∈N.

(Use the formula:

sinn x =

1

2n−1

bn/2c∑k=0

(−1)bn/2−kc(nk

)sin(n − 2k)x, n odd

12n

(n

n/2

)+

12n−1

bn/2c−1∑k=0

(−1)bn/2−kc(nk

)sin(n − 2k)x, n even

and the result of the exercise 486.)

*Ex. 490. Im,n =

∫sinm x cosn x dx, m,n ∈ Z.

(One obtains the following equivalent reduction formulas:

Im,n = −sinm−1 x cosn+1 x

n + 1+

m − 1n + 1

Im−2,n+2 =sinm+1 x cosn−1 x

m + 1+

n − 1m + 1

Im+2,n−2 =

=sinm+1 x cosn+1 x

m + 1+

m + n + 2m + 1

Im+2,n = −sinm+1 x cosn+1 x

n + 1+

m + n + 2n + 1

Im,n+2)

*Ex. 491.∫

eαx sinm βx dx, α, β ∈ R, m ∈ Z.

(Use the results of the previous exercises)

*Ex. 492.∫

eαx cosn βx dx, α, β ∈ R, n ∈ Z.

(Use the results of the previous exercises)

5.4 Determine the following indefinite integrals (primitives).

Ex. 493.∫

x2 + 2(x − 3)2(x + 2)

dx.[1925

log |x − 3| −11

5(x − 3)+

625

log |x + 2| + c]

36


Ex. 494.∫

4x − 3(x − 1)(x − 2)3 dx.

[− log |x − 1| + log |x − 2| +

2(x − 2)

−5

2(x − 2)+ c

]Ex. 495.

∫x5 + x4

− 8x3 − 4x

dx.[x3

3+

x2

2+ 4x + 2 log |x| + 5 log |x − 2| − 3 log |x + 2| + c

]Ex. 496.

∫x

(x2 + 1)(x − 1)dx.

[12

(−

12

log(x2 + 1) −12

arctan x + log |x − 1|)

+ c]

Ex. 497.∫

x + 1x2 + 1

dx.[12

log(x2 + 1) + arctan x + c]

Ex. 498.∫

x3− 6

x4 + 6x2 + 8dx.

[−

52(x − 2)2 +

1x − 2

+ log |x − 2| − log |x − 1| + c]

Ex. 499.∫

x3− 2x2 + 5

x4 + 3x3 + 3x2 − 3x − 4dx.[1

4log |x − 1| −

12

log |x + 1| +58

log |x2 + 3x + 4| −31√

78

arctan2x + 3√

7+ c

]Ex. 500.

∫2x3− 3x + 3

(x − 1)(x2 − 2x + 5)dx.[11

4log

(|x2− 2x + 5|

)+

12

log (|x − 1|) −52

arctan(14

(2x − 2))

+ 2x + c]

Ex. 501.∫

x2 + x + 1/2x2 + 1

dx.[12

log(x2 + 1) −12

arctan(x) + x + c]

Ex. 502.∫

3x2− 6x + 7

(x − 2)2(x + 5)dx.

[167

log (|x + 5|) +57

log (|x − 2|) −1

x − 2+ c

]Ex. 503.

∫2x2 + x

(x2 + 1)(x2 + 2x + 2)dx.[−

12

log(|x2 + 2x + 2|

)+

12

log(x2 + 1) + arctan(12

(2x + 2))

+ c]

Ex. 504.∫

x3 + x − 1(x2 + 2)2 dx.

[12

log(x2 + 2) −1

252

arctan(x√

2) −

x − 24x2 + 8

+ c]

Ex. 505.∫

1(x3 + 1)2 dx.[1

9log

(|x2− x + 1|

)+

29

log (|x + 1|) +2

332

arctan(2x −1√

3) +

x3x3 + 3

+ c]

Ex. 506.∫

1(x2 + 1)2 dx.

[12

arctan x +x

2x2 + 2+ c

]Ex. 507.

∫4

x4 + 1dx.[

4(

1

252

log(|x2 +

√

2x + 1|)−

1

252

log(|x2−

√

2x + 1|)

+1

232

arctan(

2x +√

2√

2

)+

1

232

arctan(

2x −√

2√

2

))+ c

]

37


Ex. 508.∫

tan2 xtan3 x + 1

dx.

[16

log(| tan2 x − tan x + 1|

)+

16

log (| tan x + 1|) −14

log(tan2 x + 1

)+

arctan(

2 tan x − 1√

3

)√

3−

12

x + c]

Ex. 509.∫

sin2 xcos2 x + 2 sin2 x

dx.[x −

arctan(√

2 tan x)

√2

+ c]

Ex. 510.∫

1sinm x cosn x

dx, m,n ∈N.[]

Ex. 511.∫

cos mx sin nx dx, m,n ∈N.[]

Ex. 512.∫ √

x4√x + 1

dx.[20

(x

14 + 1

)+

45

(x

14 + 1

)5− 5

(x

14 + 1

)4+

403

(x

14 + 1

)3− 20

(x

14 + 1

)2− 4 log

(x

14 + 1

)+ c

]Ex. 513.

∫ 3√x√

x + x2dx. []

Ex. 514.∫

1 + tan x1 − tan x

dx.[

12 log

(tan2 x + 1

)− log (| tan x − 1|) + c

]Ex. 515.

∫1

3 + 5 cos xdx.

[2(18

log(∣∣∣∣∣ sin x

cos x + 1+ 2

∣∣∣∣∣) − 18

log(∣∣∣∣∣ sin x

cos x + 1− 2

∣∣∣∣∣)) + c]

38


5.5 Definite Integrals

Determine the following definite integrals:

Ex. 516.∫ 3

−2

xx2 + 1

dx.[ log 2

2

]Ex. 517.

∫ 3

−3

xx2 + 1

dx.[0]

Ex. 518.∫ 3

−3

x2

x2 + 1dx.

[6 − 2 arctan 3

]Ex. 519.

∫ 3

−3sin3 x cos x dx.

[0]

Ex. 520.∫ 2π

0sin3 cos 2x dx.

[0]

Ex. 521.∫ π/2

π/4

xsin2 x

dx.[π

4+ log

√

2]

Ex. 522.∫ π/2

−π/2x sin x cos x dx.

[π4

]Ex. 523.

∫ π

0x sin2 x dx.

[π2

4

]Ex. 524.

∫ e

1e

x| log x|dx.[e2

4+

12−

34e2

]

Ex. 525.∫ 5

2

e2x√

ex − 1dx.

[23

((2 + e5)

√

e5 − 1 − (2 + e2)√

e2 − 1) ]

Ex. 526. Calculate∫ 10

−10f (x) dx, where:

f (x) =

x2 + 2 , x ≤ −2,√

x2 − 4x

, −2 < x < 2,√

x , x ≥ 2.

Ex. 527. Calculate∫ 10

−10f (x) dx,where:

f (x) =

1

√

x2 + 4, x ≤ 0,

x2

x2 + 1, x > 0.

39


Ex. 528. Calcolate∫ 5

−3f (x) dx,where:

f (x) =

sin

x2, x > 0,

cosx2, x < 0.

Ex. 529. Calcolate the area of the surface between the graph of the curves of equation

y = x3, y = 2 − x2 under the condition x < 0. Say if it exists (finite) the area of the surface

between the graph of the two curves, without the condition x < 0.

Ex. 530. Calcolate the area of the surface between the graph of the curves of equation

y = −x2 + x + 2 ed y = x2− 1.

40


5.6 Improper Integrals

5.6.1 Determine the convergence or divergence for the following improper integrals

Ex. 531.∫ 1

−1

1√

1 − x2dx. Calculate, if it exist, the value of the integral.

[π]

Ex. 532.∫ log 3

0

1ex − 3

dx.[Divergent

]Ex. 533. I =

∫∞

2

1x logα x

dx, α ∈ R.[If α > 1 , I =

1

(α − 1) logα−1 2; if α ≤ 1 , then I is divergent.

]Ex. 534.

∫ 6

4

1(x − 4) − log(x − 3)

dx.[Divergent

]Ex. 535.

∫ 4

2

1

|cosπx/2|3/5dx.

[Convergent

]Ex. 536.

∫ +∞

1

1((log x)(x5 + x − 2))1/5

dx.[Divergent

]Ex. 537.

∫ +∞

0

sin x log x(x + 1)3/2 − 1

dx.[Convergent

]Ex. 538.

∫ +∞

1

e1/x2− e1/x√

xdx.

[Convergent

]Ex. 539.

∫ π/2

0

e−1/x√

sin xdx.

[Convergent

]Ex. 540.

∫ +∞

0

1mx + ex dx, m ∈ R+.

[Convergent

]Ex. 541.

∫ +∞

2

1√(log x)2(x3 + x)

dx.[Convergent

]5.6.2 Discuss the integrability in an improper sense of the following integrals.

Ex. 542.∫ +∞

1

log(t + 1)t3 + 2t + 1

dt.[Convergent

]Ex. 543.

∫ 1

0

log t(1 − t)5/4t1/2

dt.[Convergent

]Ex. 544.

∫ +∞

0

1√

t(t2 + 1) log(1 +√

t)dt.

[Divergent

]

41


Ex. 545.∫ +∞

0

sin1√

y

(y − 1)1/2dy.

[Divergent

]Ex. 546.

∫ +∞

1

log(2 + x2)√

x arctan x2dx.

[Divergent

]Ex. 547.

∫ +∞

0

e−y2/2√2y + arctan(y1/4)

dy.[Convergent

]Ex. 548.

∫ +∞

1/2

e−x

(x − 3)1/3(x − 1/2)1/2dx.

[Convergent

]Ex. 549.

∫−1

+∞

e−x

(x − 4)2(x + 1/2)1/3dx.

[Divergent

]Ex. 550.

∫ +∞

1/2

1(y − 3)1/3(y − 1/2)1/2

dy.[Divergent

]Ex. 551.

∫ +∞

1/2

1|x − 3|3/4(x − 1/2)1/2

dx.[Convergent

]Ex. 552.

∫ +∞

3

log(3 + x−1/4)(x − 3)3/4(x − 1/2)1/2

dx.[Convergent

]Ex. 553.

∫ 1

0

log x2

(1 − x)9/4x1/2dx.

[Divergent

]Ex. 554. If Ia =

∫ +∞

a

e−x

(x − 3)2(x − 1/2)1/2dx, find a ∈ R such that Ia < +∞.

[a > 3

]Ex. 555. If Ia =

∫ +∞

1

dy(1 + y)2(y + 2)a dy, find a ∈ R such that Ia < +∞.

Moreover, calculate I1.[a > −1. I1 =

12

+ log23

]5.6.3 Determine the values of α ∈ R s.t. the following improper integrals result to be

convergent.

Ex. 556.∫ 1

0

(tan x)α

log(1 + sin x)dx.

[α > 0

]Ex. 557.

∫ +∞

0

arctan(1/xα)√

x + 2dx.

[α >

12

]Ex. 558.

∫ 1

0

cos x + 3xα +

√x

dx.[α < 1

]Ex. 559.

∫ +∞

2

arctan(x + 7)x logα(x − 2)

dx.[α > 1

]

42


Ex. 560.∫ +∞

2

logα(1 + 1/x)√

x + 1dx.

[α >

12

]Ex. 561.

∫ +∞

1

|sin(1/x) − 1/x|α2/2

3√xdx.

[|α| >

23

]Ex. 562.

∫ +∞

1

(1 − cos

1x3

)αxα/2 dx.

[α >

211

]Ex. 563.

∫ +∞

0(arctan x)α(

√x + 3)2α dx. [Divergent for any α]

Ex. 564.∫ +∞

0(e−x +

x2α + 1√

x) dx. [Divergent for any α]

Ex. 565.∫ +∞

−1

arctan(x2 + 3)(x + 1)α(x + 2)

dx.[0 < α < 1

]Ex. 566.

∫ +∞

0arctan(1/x)α(x2 + 3)2α dx.

[−

14< α < 0

]Ex. 567.

∫ +∞

3

e−t

(t − 3)α√

tdt.

[α < 1

]Ex. 568.

∫ +∞

0

sinα(1/√

t)√

t logα(t + 1)dt. [Divergent for any α]

Ex. 569.∫ 2

−1

(ex+3 + 7 sin2 x)xα(ex + 1)

dx.[α < 1

]Ex. 570.

∫ +∞

−∞

e−αx2/2 dx.[α > 0

]Ex. 571.

∫ +∞

1(e1/x

− 1)αlog(2 + x)

x2 dx.[α > −1

]Ex. 572.

∫ +∞

4

logα+1(x − 3)√

ex−4 − 1dx. Moreover, calculate its value for α = −1.

[α > −

32. π

]

Ex. 573.∫ +∞

0

sin( xx2 + 1

)(x2 − sin x2)α

dx.[0 < α <

13

]Ex. 574.

∫ +∞

0

3 + 2 sin x(x − 1)1/3(x + 2)4α

dx.[α >

16

]Ex. 575.

∫ +∞

0

log(1 + xα)x3 dx.

[α > 2

]Ex. 576.

∫ 1

0

1x(− log x)α + x2(1 − x2)1/3

dx.[α > 1

]

43


5.6.4 Determine the values of α and β such that the following improper integrals

converge

Ex. 577.∫ 1

0

| log x|α

| sinπx|βdx.

[β < 1, β − α < 1

]Ex. 578.

∫ +∞

0

eαx+β/x

x + 1dx.

[α < 0, β ≤ 0

]Ex. 579.

∫ +∞

0

(arctan x)α

xβ(2 + cos x)dx.

[β > 1, β − α < 1

]5.7 Numerical Series

5.7.1 Determine the nature of the following numerical series

Ex. 580.∞∑

k=1

1

k +√

k.

[Divergent

]

Ex. 581.∞∑

k=1

kk + log k

.[Divergent

]

Ex. 582.∞∑

k=1

1klog k

.[Convergent

]

Ex. 583.∞∑

k=1

(log(log k)

log k

)k

.[Convergent

]

Ex. 584.∞∑

k=1

(k!)2

(2k)!.

[Convergent

]

Ex. 585.∞∑

k=1

k2e−√

k.[Convergent

]

Ex. 586.∞∑

k=1

(√k2 + 1 − k

)log

(1 +

1k

).

[Convergent

]

Ex. 587.∞∑

k=1

(√k + 1 −

√

k)2.

[Divergent

]

Ex. 588.∞∑

k=1

√

1 + sin3k− 1

(1 − e−1/k).

[Convergent

]

Ex. 589.∞∑

k=1

(e1/k2− 2 cos

1k

+ 1).[Convergent

]

44


Ex. 590.∞∑

k=1

13 + eαk

, α ∈ R.[Convergent if α > 0, divergent otherwise

]

Ex. 591.∞∑

k=1

k2

4 + eαk, ∈ R.

[Convergent if α > 0, divergent otherwise

]

Ex. 592.∞∑

k=1

1 −√

e(cos

1k

)k2 . [Convergent

]

Ex. 593.∞∑

k=1

( 59 − 2 cos k

)k.

[Convergent

]

Ex. 594.∞∑

k=1

(log(1 +

33√

k2) −

α3√

k2

).

[Convergent if α = 3, divergent otherwise

]

Ex. 595.∞∑

k=1

(k sin

(1k

))k3

.[Convergent

]5.7.2 Determine the nature of the following series

Ex. 596.∞∑

k=1

(1 −

1k1/3

)k2

.[Convergent

]

Ex. 597.∞∑

k=1

( 35 + cos2 k

)k.

[Convergent

]

Ex. 598.∞∑

k=1

(63n +

(−1)n+1

4n

). Moreover, if it exists, calculate the sum. [Converges to

233

]

Ex. 599.∞∑

n=1

n√

n

en2 .[Convergent

]

Ex. 600.∞∑

k=1

(3x2− 3

x2 + 1

)2n

+n + 1

n2(log n)x + 2

, x ∈ R.[Converges if1 < x <

√2, diverges otherwise

]5.7.3 Determine the nature of the following series for any α ∈ R and x ∈ R

Ex. 601.∞∑

k=4

1k2

(1 −

1k

)kα

.[Converges for any α ∈ R

]

Ex. 602.∞∑

n=1

nα(x + 1)2n

(2n)!, x ∈ R.

[Converges for any α, x ∈ R

]

45


Ex. 603.∞∑

n=1

n(1 −

(1 +

1n2α

)1/4).

[Converges if α > 1, divergent otherwise

]

Ex. 604.∞∑

n=1

n8

(n − log n)10 − nα.

[Converges if α , 10, divergent otherwise

]

Ex. 605.∞∑

n=1

nα((

n4− 5n2

)1/4−

(n3− 3n

)1/3).

[Convergent if α < 0, divergent otherwise

]Ex. 606. Determine the values of α ∈ R such that the following two series have the same

nature:∞∑

n=1

(e(nα+1/n)

− 1),

∞∑n=1

log(1 + nα).[α ≥ −1

]

Ex. 607. Study the nature of the series∞∑

n=1

3(−1)n3αn for any α ∈ R. Moreover, calculate its

sum once calculated the one of the two series∞∑

n=1

3 · 32αn and∞∑

n=1

13· 3(2n+1)α.[

Converges for α < 0 to the value9 + 3α

3(1 − 32α)

]5.7.4 Discuss the simply and absolute convergence of the following series

Ex. 608.∞∑

n=0

arctan1

n + 1.

[Simply and absolutely convergent

]

Ex. 609.∞∑

n=0

(α

2α + 3

)n 1log n

.[Simply convergent if α < −3, absolutely convergent if α < −3, α ≥ −1

]Ex. 610.

∞∑k=1

(−1)k(e1/k1/4

− 1)α.

[Simply convergent if α > 0, absolutely convergent if α > 4

]

46

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Exercises in Mathematical Analysis I - uniroma2.itciolli/ExAn1.pdf · Università di Tor Vergata – Dipartimento di Ingegneria Civile ed Ingegneria Informatica Exercises in Mathematical