- Topics At a Glance
- Indefinite Integrals Introduction
- Integration by Substitution: Indefinite Integrals
- Legrange (Prime) Notation
- Leibniz (Fraction) Notation
- Integration by Substitution: Definite Integrals
**Integration by Parts: Indefinite Integrals****Some Tricks**- Integration by Parts: Definite Integrals
- Integration by Partial Fractions
- Integrating Definite Integrals
- Choosing an Integration Method
- Integration by Substitution
- Integration by Parts
- Integration by Partial Fractions
- Thinking Backwards
- Improper Integrals
- Badly Behaved Limits
- Badly Behaved Functions
- Badly Behaved Everything
- Comparing Improper Integrals
- The
*p*-Test - Finite and Infinite Areas
- Comparison with Formulas
- In the Real World
- I Like Abstract Stuff; Why Should I Care?
- How to Solve a Math Problem

We can use integration by parts to find the integral of something that doesn't look like a product. This is because whatever the integrand is, we can think of it as the product of itself and 1. Then we can choose *v'* = 1 and apply the integration-by-parts formula.

For example, since

*ln* *x* = (*ln* *x*)(1),

we know

If we chose *u* = 1 then *u'* would be zero, which doesn't seem like a good idea. So take

*u* = *ln* *x*

*v'* = 1

Sometimes we need to rearrange the integrand in order to see what *u* and *v*' should be. Exponents can be deceiving.

For example, look at the integral

This looks like a product, so we want to use integration by parts. However, choosing

*u* = *x*^{5}

or

*u* = *e*^{x3}

won't work very well (try it yourself if you don't believe us; we're not going to demonstrate). But what if we rewrite the integrand by factoring *x*^{5}?

Now we can see it's reasonable to choose

*v'* = *x*^{2}*e*^{x3},

since we can use substitution to figure out the antiderivative *v*. This leaves

*u* = *x*^{3}.

We already did some exercises where you had to integrate by parts twice: once to start off, then again to find the new integral. There are some problems where, if you integrate by parts twice, the original integral shows up again. Sometimes this will be as helpful as the equation

0 = 0

(that is, not helpful at all). However, sometimes when the original integral shows up again, you'll get an equation that you can rearrange to solve for the original integral.

Exercise 1

Apply the formula for integration by parts with *u* = *ln* x and *v'* = 1 to find

Exercise 2

Use integration by parts to find .

Exercise 3

Find

by applying the formula for integration by parts with *u* = *x*^{3} and *v'* = *x*^{2}*e*^{x3}.

Exercise 4

Find

Exercise 5

Use integration by parts to find

taking *u* = sin *x* for the first integration.

Exercise 6

Integrate.

Exercise 7

Integrate.

Exercise 8

Integrate.

Exercise 9

Integrate.

Exercise 10

Integrate.

Exercise 11

Integrate.

Exercise 12

Integrate.