Integrals

ByZac CockmanLiz Mooney

Integration Techniques

• Integration is the process of finding an indefinite or diefinite integral

• Integral is the definite integral is the fundamental concept of the integral calculus. It is written as

• Where f(x) is the integrand, a and b are the lower and upper limits of integration, and x is the variable of integration.

Integration techniques

• Integration is the opposite of Differentiation.

• Power Rule• U-Substitution• Special Cases• Sin and Cos

Power Rule

•

• n cannot equal -1• u=x• Du=dx• N=1

• C = constant

+ c

Examples

U = 2xDu = dxn = 1

Answer = X2 + C

Examples

• Answer = X3/3 + 3x2/2 + 2x + C

U=x U=x U=x

Du=dx Du=dx Du=dx

N=2 N=1 N=0

Examples

• U = 4 X2

• Du = 8xdx• N = -1/2

• 3/8 * 2 * (4x2 + 5)1/2 + C• Answer ¾(4X2 + 5)1/2 + C

Try Me

Try Me

•

Try Me Continue

• U = 1 +x2

• Du = 2dx• N = -1/2• 1/2 [2(1+x2)1/2] + C• (1+x2)1/2 + C

Try Me

Try Me

• U = x4 + 3• Du = 4x3 dx• N = 2

Try Me Continued

U-Sub

• What is U-Sub• When do you use it• Steps• Find your u, du, and for u, solve for x• Replace all the x for u. • Do the same steps for power rule• At the end replace the u in the problem for

your u when you found it in the beginning.

Example

• U=• X= u2 -1• dx= 2udu• (u2 – 1) u(2udu)• 2u4 – 2u2

• 2/5 (u5 – 2/3u3) + c• 2/5 (x+1) 5/2 + c

Example

• U =• U2 – 1 = x• 2udu = dx

2/3(x+1)3/2 -2(x+1) + c

Try Me

Try Me

U =

X =

Du = udu

1/10 u5 + 1/2u2 + c

1/10 (2x-3)5/2 + ½(2x-3) + c

Special Cases

• When n = -1 the u is put inside the absolute value of the natural log

• If there is only one x in the problem and it is squared, square the term before taking the interval

Special Cases

• Examples

• U = x-1• Du = dx• N = -1

Special Cases

• Examples

Integration using Powers of Sin and Cos

• Three Methods– Odd-Even Odd-Odd Even-Even

• In Odd-Even, take the odd power and re write the odd power as odd even

• Re write the even power change it using Pythagorean identity.

• In Odd-Odd, take one of the odds, change to odd even

• Use same rules

Integration using Powers of sin and cos

• For Even-Even, change the power to the half angle formula.

Special CaseIf the Power of the trig is 1, u is the angle

Powers of Trig Odd - Even

• Take the odd power, re write the odd power as odd

even

• Re write the even power, change it using the

Pythagorean identities.

• ∫sin5xcos4xdx

• ∫sin4x sinxcos4xdx

• ∫(1-cos2x)2 sinxcos2xdx

Powers of Trig Odd-Even

• ∫(1-2cos2x+cos4x) sinxcos4xdx

• ∫sinxcos4xdx-2 ∫sinx cos6xdx+∫sinxcos8xdxU = cosx U = cosx U = cosx

Du = -sinx Du = -sinx Du = -sinx

N = 4 N = 6 N = 8

-1/5cos5x+2/7cos7x-1/9cos9x+c

Try Me

∫sin32xcos22xdx

Powers of Trig Odd - Even

• Try Me• ∫sin32xcos22xdx• ∫sin22xsin2xcos22xdx• ∫(1-cos22x) sin2xcos22xdx• -1/2 ∫sin22xcos22xdx+1/2 ∫sin2xcos42xdx

U = cosx U = cosx

Du = -2sin2x Du = -2sin2x

N = 2 N = 4

-1/6 cos32x+1/10cos52x+c

Powers of Trig Odd Odd

• Take one of the odds, change to odd even. Use other rules to finish.

• Example

Powers of trig Odd-Odd

U = cosx U = cosx

du= -sinxdx du= -sinxdx

n = 3 n = 5

Powers of Trig Odd-Odd

• Example

Powers of Trig Odd Odd

• Example Continued

U = cosx U = cosx

Du = -sinx Du = -sinx

N = 17 N = 19

Powers of Trig Even-Even

• Change to half angle formula• ∫sin2xdx• ∫1-cos2xdx• 2• 1/2∫dx-(1/2)(1/2)∫2cos2xdx

U = x U = 2x

Du = dx Du = 2dx

1/2x-1/4sin2x+c

Try Me

∫sin2xcos2x

Powers of Trig Even-Even

• Try Me• ∫sin2xcos2x• ∫(1-cos2x)(1+cos2x)

2 2• 1/4∫(1-cos22x)dx• 1/4∫sin22xdx• 1/4∫1-cos4x/2dx• 1/8∫dx-(1/4)(1/8) ∫4cos4xdx• 1/8x-1/32sin4x+c

Solving for Integrals

• U =x-1• Du = dx• N = 2

• 9 – 0 = 9

Try Me

• Try Me

Solving for Integrals

• U = x2 + 2• Du = 2xdx• N = 2

Bibliography

• www.musopen.com• Mathematics Dictionary, Fourth Edition,

James/James, Van Nostrand Reinnhold Company Inc., 1976