- 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

You've been learning the different methods of integration in a very artificial environment. You know that if you're in the "Integration by Substitution" section, you use substitution. If you're in the "Integration by Parts" section, you use integration by parts.

In the real world (by which we mean on exams), the directions probably won't say which method to use—you'll have to figure that out yourself.

The more you practice, the better you'll get at figuring out which method to use - you won't even have to think about it. In the meantime, we have some hints.

Exercise 1

For the integral, a. Determine whether the integral should be integrated by substitution, parts, partial fractions, or thinking backwards (no fancy techniques required).b. If you said 'substitution' for part (a), identify *u*. If you said `parts', identify *u* and *v*'. If you said `thinking backwards', go ahead and find the integral.

Exercise 2

For the integral, a. Determine whether the integral should be integrated by substitution, parts, partial fractions, or thinking backwards (no fancy techniques required).b. If you said 'substitution' for part (a), identify *u*. If you said `parts', identify *u* and *v*'. If you said `thinking backwards', go ahead and find the integral.

Exercise 3

For the integral, a. Determine whether the integral should be integrated by substitution, parts, partial fractions, or thinking backwards (no fancy techniques required).b. If you said 'substitution' for part (a), identify *u*. If you said `parts', identify *u* and *v*'. If you said `thinking backwards', go ahead and find the integral.

Exercise 4

*u*. If you said `parts', identify *u* and *v*'. If you said `thinking backwards', go ahead and find the integral.

Exercise 5

*u*. If you said `parts', identify *u* and *v*'. If you said `thinking backwards', go ahead and find the integral.

Exercise 6

*u*. If you said `parts', identify *u* and *v*'. If you said `thinking backwards', go ahead and find the integral.

Exercise 7

*u*. If you said `parts', identify *u* and *v*'. If you said `thinking backwards', go ahead and find the integral.

Exercise 8

*u*. If you said `parts', identify *u* and *v*'. If you said `thinking backwards', go ahead and find the integral.

Exercise 9

*u*. If you said `parts', identify *u* and *v*'. If you said `thinking backwards', go ahead and find the integral.

Exercise 10

*u*. If you said `parts', identify *u* and *v*'. If you said `thinking backwards', go ahead and find the integral.

Exercise 11

*u*. If you said `parts', identify *u* and *v*'. If you said `thinking backwards', go ahead and find the integral.

Exercise 12

*u*. If you said `parts', identify *u* and *v*'. If you said `thinking backwards', go ahead and find the integral.

Exercise 13

*u*. If you said `parts', identify *u* and *v*'. If you said `thinking backwards', go ahead and find the integral.

Exercise 14

*u*. If you said `parts', identify *u* and *v*'. If you said `thinking backwards', go ahead and find the integral.

Exercise 15

*u*. If you said `parts', identify *u* and *v*'. If you said `thinking backwards', go ahead and find the integral.

Exercise 16

*u*. If you said `parts', identify *u* and *v*'. If you said `thinking backwards', go ahead and find the integral.

Exercise 17

*u*. If you said `parts', identify *u* and *v*'. If you said `thinking backwards', go ahead and find the integral.

Exercise 18

*u*. If you said `parts', identify *u* and *v*'. If you said `thinking backwards', go ahead and find the integral.

Exercise 19

*u*. If you said `parts', identify *u* and *v*'. If you said `thinking backwards', go ahead and find the integral.

Exercise 20

*u*. If you said `parts', identify *u* and *v*'. If you said `thinking backwards', go ahead and find the integral.

Exercise 21

*u*. If you said `parts', identify *u* and *v*'. If you said `thinking backwards', go ahead and find the integral.

Exercise 22

*u*. If you said `parts', identify *u* and *v*'. If you said `thinking backwards', go ahead and find the integral.