Improper Integrals Examples: 43 Problems with Answers

Start with the recap, study the fully worked examples, then use the practice problems to check your understanding of Improper Integrals.

This page combines explanation, solved examples, and follow-up practice so you can move from recognition to confident problem-solving in Math.

Concept Recap

Integrals where the interval of integration is infinite (Type I: $\int_a^{\infty} f(x)\,dx$ ) or the integrand has an infinite discontinuity on the interval (Type II: $\int_a^b f(x)\,dx$ where $f$ blows up at some point in $[a, b]$ ). Evaluated as limits of proper integrals.

Can an infinite region have a finite area? Surprisingly, yes. The area under $\frac{1}{x^2}$ from 1 to infinity is exactly 1. Improper integrals extend integration to infinite intervals and unbounded functions by using limits to handle the 'improper' part.

Read the full concept explanation →

How to Use These Examples

Read the first worked example with the solution open so the structure is clear.
Try the practice problems before revealing each solution.
Use the related concepts and background knowledge badges if you feel stuck.

What to Focus On

Core idea: Integrate over an infinite interval or through a blow-up by taking a limit of ordinary integrals.

Common stuck point: The procedure for improper integrals is the easy part; the trap is treating $\infty$ as a number to plug in. Asking "Does this integral run to infinity or pass through a point where the integrand blows up?" first is what keeps a correct-looking calculation from being attached to the wrong concept.

Sense of Study hint: Ask: Does this integral run to infinity or pass through a point where the integrand blows up?

Worked Examples

Example 1

easy

Evaluate

\displaystyle\int_1^{\infty} \frac{1}{x^2}\,dx

Answer

1

First step

Replace the infinite upper limit with a variable:

\int_{1}^{\infty} \frac{1}{x^{2}} d x = \lim_{b \to \infty} \int_{1}^{b} x^{- 2} d x

Full solution

2
Integrate $x^{-2}$ : $= \lim_{b \to \infty} {[- \frac{1}{x}]}_{1}^{b} = \lim_{b \to \infty} (- \frac{1}{b} + 1)$
3
Take the limit as $b \to \infty$ : since $\frac{1}{b} \to 0$ , the integral converges to $1$ .

Replace

\infty

with

b

, integrate, take the limit. The

1/b

term vanishes.

Example 2

hard

Evaluate

\displaystyle\int_0^1 \frac{1}{\sqrt{x}}\,dx

(Type II).

Example 3

medium

Evaluate

\int_0^{\infty}xe^{-2x}\,dx

Example 4

medium

Evaluate

\int_0^{\infty}\frac{1}{(1+x)^2}\,dx

Example 5

hard

Evaluate

\int_0^{\infty}x^2e^{-x}\,dx

Example 6

hard

Evaluate

\int_1^{\infty}\frac{1}{x^2(x+1)}\,dx

Example 7

challenge

Evaluate

\int_0^1\frac{\ln x}{\sqrt{x}}\,dx

Practice Problems

Try these problems on your own first, then open the solution to compare your method.

Example 1

easy

Does

\displaystyle\int_1^{\infty} \frac{1}{x}\,dx

converge or diverge?

Example 2

medium

Evaluate

\displaystyle\int_0^{\infty} e^{-x}\,dx

Example 3

easy

Why is

\int_1^{\infty}\frac{1}{x^2}\,dx

improper?

Example 4

easy

Why is

\int_0^1\frac{1}{\sqrt{x}}\,dx

improper?

Example 5

easy

Evaluate

\int_1^{\infty}\frac{1}{x^2}\,dx

Example 6

easy

Does

\int_1^{\infty}\frac{1}{x}\,dx

converge or diverge?

Example 7

easy

For the

p

-integral

\int_1^{\infty}\frac{1}{x^p}\,dx

, when does it converge?

Example 8

easy

For

\int_0^1\frac{1}{x^p}\,dx

, when does it converge?

Example 9

easy

Rewrite

\int_a^{\infty}f(x)\,dx

using a limit.

Example 10

easy

Evaluate

\int_0^1\frac{1}{\sqrt{x}}\,dx

Example 11

medium

Evaluate

\int_0^{\infty}e^{-x}\,dx

Example 12

medium

Evaluate

\int_1^{\infty}\frac{1}{x^3}\,dx

Example 13

medium

Evaluate

\int_0^{\infty}\frac{1}{1+x^2}\,dx

Example 14

medium

Use comparison to decide if

\int_1^{\infty}\frac{1}{x^2+1}\,dx

converges.

Example 15

medium

Use comparison to decide if

\int_1^{\infty}\frac{1}{\sqrt{x}+1}\,dx

converges.

Example 16

medium

Evaluate

\int_0^1\ln x\,dx

Example 17

medium

Evaluate

\int_2^{\infty}\frac{1}{x\ln x}\,dx

Example 18

medium

Evaluate

\int_{-\infty}^{\infty}\frac{1}{1+x^2}\,dx

Example 19

medium

Evaluate

\int_0^{\infty}e^{-2x}\,dx

Example 20

challenge

Evaluate

\int_0^{\infty}xe^{-x}\,dx

Example 21

challenge

Find all

p

for which

\int_0^{\infty}\frac{1}{x^p}\,dx

converges.

Example 22

challenge

Evaluate

\int_1^{\infty}\frac{\ln x}{x^2}\,dx

Example 23

easy

Does

\int_1^{\infty}\frac{1}{\sqrt{x}}\,dx

converge or diverge?

Example 24

easy

Which is the type of improperness in

\int_3^{\infty}\frac{1}{x^2+1}\,dx

Example 25

easy

Which is the type of improperness in

\int_0^4\frac{1}{\sqrt{x}}\,dx

Example 26

easy

Does

\int_0^1\frac{1}{x^2}\,dx

converge or diverge?

Example 27

medium

Evaluate

\int_{-\infty}^0 e^x\,dx

Example 28

medium

Use the

p

-test to classify

\int_0^1\frac{1}{x^{2/3}}\,dx

Example 29

medium

Use comparison to decide whether

\int_1^{\infty}\frac{\sin^2 x}{x^2}\,dx

converges.

Example 30

medium

Evaluate

\int_0^{\infty}\frac{2}{(x+1)^3}\,dx

Example 31

medium

Evaluate

\int_0^1\frac{1}{\sqrt[3]{x}}\,dx

Example 32

medium

Evaluate

\int_1^{\infty}\frac{1}{x(x+1)}\,dx

Example 33

hard

Evaluate

\int_0^{\infty}\frac{x}{(1+x^2)^2}\,dx

Example 34

hard

Use comparison to show

\int_1^{\infty}\frac{1}{x^2+\sqrt{x}}\,dx

converges.

Example 35

hard

Evaluate

\int_0^{\infty}\frac{1}{e^x+e^{-x}}\,dx

Example 36

challenge

Evaluate

\int_0^{\infty}\frac{\arctan x}{1+x^2}\,dx

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Related Concepts

Definite Integral Limit Infinity Convergence and Divergence

Background Knowledge

These ideas may be useful before you work through the harder examples.

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