- 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 figure out whether integrals converge or diverge by comparing them with other integrals whose convergence or divergence we already know. When we're looking at formulas and not at graphs, we have to figure out from scratch what to compare an integral to.

- If we want to show that an integral converges, we have to find a larger function whose integral on the same interval converges.
- If we want to show that an integral diverges, we have to find a smaller function whose integral on the same interval diverges.

Example 1

Let converge or diverge? |

Example 2

Let . Does converge or diverge? |

Example 3

Does converge or diverge? |

Example 4

Does converge or diverge? |

Exercise 1

Determine if the integral converges or diverges. What integral are you using for comparison in each case?

Exercise 2

Determine if the integral converges or diverges. What integral are you using for comparison in each case?

Exercise 3

Determine if the integral converges or diverges. What integral are you using for comparison in each case?

Exercise 4

(hint: for *x* > 1, we know sin *x* < *x* and )

Exercise 5

Exercise 6

Exercise 7