Comparison Theorem for Improper Integrals

Comparison Theorem for Improper Integrals

Common Improper Integrals

1. ${\displaystyle\int^\infty_a}\frac{1}{x^p}\de{x}$

• If $p\le1$: diverges
• If $p>1$: converges

2. ${\displaystyle\int^a_0}\frac{1}{x^p}\de{x}$

• If $p<1$: converges
• If $p\ge1$: diverges

Comparison Theorem

Suppose that $f$ and $g$ are continuous and $f(x)\ge g(x)\ge 0$ for $x \ge a.$

a. If ${\displaystyle\int^\infty_a}f(x)\de{x}$ converges, then ${\displaystyle\int^\infty_a}g(x)\de{x}$ converges.

b. If ${\displaystyle\int^\infty_a}g(x)\de{x}$ diverges, then ${\displaystyle\int^\infty_a}f(x)\de{x}$ diverges.

Notes:

• The theorem only tells you if an improper integral diverges or converges, not the value it converges to
• Often, we will use the common improper integrals and the Comparison Theorem