MATH148 c Justin Cantu

Section 7.4: Improper Integrals

R b When defining a definite integral a f(x) dx we dealt with a function f defined on a finite closed interval [a, b] and we assumed that f does not have an infinite discontinuity on [a, b]. We will relax these restrictions and extend the concept of a definite integral to the cases where

1. One or both limits of integration are infinite (unbounded intervals),

2. The integrand f becomes infinite at one or more points of the interval of integration (infinite discontinuities).

Such definite integrals are called improper integrals.

Type I: Unbounded Intervals: One or both limits of integration are infinite.

Example: Suppose we want to compute the area of the unbounded region below the graph of 1 f(x) = and above the x-axis for x ≥ 1. x2

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For functions that are continuous on unbounded intervals, we define Z ∞ Z z f(x) dx = lim f(x) dx a z→∞ a and Z b Z b f(x) dx = lim f(x) dx −∞ z→−∞ z These improper integrals are called convergent if the corresponding limit exists and divergent if the limit does not exist.

Z ∞ 1 Example: √ dx 1 x

We have the following general result related to the last two examples. The p-test for Improper Integrals: For a > 0, the improper integral

Z ∞ 1 p dx a x converges if p > 1 and diverges if p ≤ 1.

Z 2 Example: 3x2ex3 dx −∞

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If f(x) is continuous on (−∞, ∞), then

Z ∞ Z a Z ∞ f(x) dx = f(x) dx + f(x) dx −∞ −∞ a for any real number a. If both improper integrals on the right converge, then the left improper integral converges to their sum. Otherwise, the improper integral on the left diverges. Example: Compute the following improper integrals.

Z ∞ 1 (a) 2 dx −∞ 1 + x

Z ∞ (b) x3 dx −∞

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Type II: Unbounded Integrand The integrand has an infinite discontinuity in the interval of integration. 1 Example: Suppose we want to find the area of the unbounded region between f(x) = √ and the x x-axis on the interval [0, 1].

If f is continuous on (a, b] and limx→a+ f(x) = ±∞, we define

Z b Z b f(x) dx = lim f(x) dx + a c→a c

If f is continuous on [a, b) and limx→b− f(x) = ±∞, we define

Z b Z c f(x) dx = lim f(x) dx − a c→b a

Z 0 1 Example: 2 dx −1 x

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Comparison Test for Improper Integrals:

In many cases, it is impossible to evaluate an improper integral exactly, but it is still helpful to know whether it is convergent or divergent. We can determine convergence/divergence by comparing to an improper integral whose behavior is easier to analyze.

Suppose that f and g are continuous functions with 0 ≤ f(x) ≤ g(x) on [a, ∞]. R ∞ R ∞ 1. If a g(x) dx is convergent, then a f(x) dx is convergent. R ∞ R ∞ 2. If a f(x) dx is divergent, then a g(x) dx is divergent.

R ∞ R ∞ Note: If a g(x) dx is divergent or a f(x) dx is convergent, then no conclusion can be made.

Example: Determine whether the following integrals converge or diverge.

Z ∞ 1 1. 4 dx 1 x + 3

Z ∞ 1 + e−x 2. dx 1 x

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Z ∞ 1 3. x dx 1 x + e

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