Review the following concepts if needed:
 Integration by Parts Help
 Partial Fractions Help
 Methods of Integration  Substitution Help
 Integrals of Trigonometric Expressions Help
Methods Of Integration Practice Test

Use integration by parts to evaluate each of the following indefinite integrals.
(a) ∫ log^{2} x dx
(b) ∫ x · e^{3x} dx
(c) ∫ x ^{2} cos x dx
(d) ∫ t sin 3 t cos 3 t dt
(e) ∫ cos y ln(sin y ) dy
(f) ∫ x^{2} e^{4 x} dx

Use partial fractions to evaluate each of the following indefinite integrals.

Use the method of u substitution to evaluate each of the following indefinite integrals.

Evaluate each of the following indefinite trigonometric integrals.
(a) ∫ sin x cos^{2} x dx
(b) ∫ sin^{3} x cos^{2} x dx
(c) ∫ tan^{3} x sec^{2} x dx
(d) ∫ tan x sec^{3} x dx
(e) ∫ sin^{2} x cos^{2} x dx
(f) ∫ sin x cos^{4} x dx

Calculate each of the following definite integrals.
Answers

We do (a), (b), (c), (d).
(d) Notice that . Now let u = t and dv = sin 6 t d t . Then

We do (a), (b), (c), (d).

We do (a), (b), (c), (d).
(a) Let u = sin x, du = cos x dx . Then the integral becomes
Resubstituting x , we obtain the final answer
(b) Let . Then the integral becomes
∫ 2 sin u du = −2 cos u + C.
Resubstituting x , we obtain the final answer
(c) Let u = ln x, du = [1/ x ] dx . Then the integral becomes
Resubstituting x , we obtain the final answer
(d) Let u = tan x, du = sec ^{2} x dx. Then the integral becomes
∫ e ^{u} du = eu + C.
Resubstituting x, we obtain the final answer
∫ e ^{tan x} sec ^{2} x dx = e ^{tan x} + C.

We do (a), (b), (c), (d).
(a) Let u = cos x, du = − sin x dx. Then the integral becomes
Resubstituting x , we obtain the final answer
(b) Write
Let u = cos x, du = − sin x dx . Then the integral becomes
Resubstituting x , we obtain the final answer
(c) Let u = tan x, du = sec ^{2} x dx . Then the integral becomes
Resubstituting x , we obtain the final answer
(d) Let u = sec x, du = sec x tan x . Then the integral becomes
Resubstituting x , we obtain the final answer

We do (a), (b), (c), (d).
(a) Use integration by parts twice:
We may now solve for the desired integral:
(b) Integrate by parts with u = ln x, dv = x ^{2} dx . Thus
(c) We write
Thus
(d) We write
Ask a Question
Have questions about this article or topic? AskRelated Questions
Q:
Q:
Q:
Q:
Popular Articles
 Kindergarten Sight Words List
 First Grade Sight Words List
 10 Fun Activities for Children with Autism
 Grammar Lesson: Complete and Simple Predicates
 Definitions of Social Studies
 Child Development Theories
 Signs Your Child Might Have Asperger's Syndrome
 How to Practice Preschool Letter and Name Writing
 Social Cognitive Theory
 Curriculum Definition