- 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

As with integration by substitution, there are two distinct ways to integrate definite integrals using integration by parts. As with integration by substitution, we have to be careful not to mix the ways up!

**Way 1: First integrate the indefinite integral**

This method is analogous to the Way 1 we introduced for integration by substitution. When asked to evaluate a definite integral,

- use integration by parts to find an antiderivative of the integrand and then

- use the FTC to evaluate the integral.

**Way 2: Keep the limits of integration after applying the formula**

The second way to use integration by parts on definite integrals is to keep the limits of integration all the way through. When we apply the magic formula for integration by parts, it will look like this:

We're not just keeping the limits of integration on the integral; we also need to evaluate *uv* from *a* to *b*:

This is the step that's easiest to forget, so we'll probably remind you several more times.

Example 1

Find |

Example 2

Find |

Exercise 1

Evaluate the definite integral using integration by parts with Way 1.

Exercise 2

Evaluate the definite integral using integration by parts with Way 1.

Exercise 3

Evaluate the definite integral using integration by parts with Way 1.

Exercise 4

Evaluate the definite integral using integration by parts with Way 1.

Exercise 5

Evaluate the definite integral using integration by parts with Way 1.

Exercise 6

Evaluate the definite integral using integration by parts with Way 2.

Exercise 7

Evaluate the definite integral using integration by parts with Way 2.

Exercise 8

Evaluate the definite integral using integration by parts with Way 2.

Exercise 9

Evaluate the definite integral using integration by parts with Way 2.

Exercise 10

Evaluate the definite integral using integration by parts with Way 2.