Improper Integrals Examples in Math

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

Can an infinite region have a finite area? Surprisingly, yes. The area under 1x2\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 ∫1∞1x2 dx\displaystyle\int_1^{\infty} \frac{1}{x^2}\,dx.

Answer

11

First step

1
Replace the infinite upper limit with a variable: ∫1∞1x2 dx=lim⁡b→∞∫1bx−2 dx\int_1^{\infty}\frac{1}{x^2}\,dx = \lim_{b\to\infty}\int_1^b x^{-2}\,dx

Full solution

  1. 2
    Integrate x−2x^{-2}: =lim⁡b→∞[−1x]1b=lim⁡b→∞(−1b+1)= \lim_{b\to\infty}\left[-\frac{1}{x}\right]_1^b = \lim_{b\to\infty}\left(-\frac{1}{b} + 1\right)
  2. 3
    Take the limit as b→∞b \to \infty: since 1b→0\frac{1}{b} \to 0, the integral converges to 11.
Replace ∞\infty with bb, integrate, take the limit. The 1/b1/b term vanishes.

Example 2

hard
Evaluate ∫011x dx\displaystyle\int_0^1 \frac{1}{\sqrt{x}}\,dx (Type II).

Example 3

medium
Evaluate ∫0∞xe−2x dx\int_0^{\infty}xe^{-2x}\,dx.

Example 4

medium
Evaluate ∫0∞1(1+x)2 dx\int_0^{\infty}\frac{1}{(1+x)^2}\,dx.

Example 5

hard
Evaluate ∫0∞x2e−x dx\int_0^{\infty}x^2e^{-x}\,dx.

Example 6

hard
Evaluate ∫1∞1x2(x+1) dx\int_1^{\infty}\frac{1}{x^2(x+1)}\,dx.

Example 7

challenge
Evaluate ∫01ln⁡xx dx\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 ∫1∞1x dx\displaystyle\int_1^{\infty} \frac{1}{x}\,dx converge or diverge?

Example 2

medium
Evaluate ∫0∞e−x dx\displaystyle\int_0^{\infty} e^{-x}\,dx.

Example 3

easy
Why is ∫1∞1x2 dx\int_1^{\infty}\frac{1}{x^2}\,dx improper?

Example 4

easy
Why is ∫011x dx\int_0^1\frac{1}{\sqrt{x}}\,dx improper?

Example 5

easy
Evaluate ∫1∞1x2 dx\int_1^{\infty}\frac{1}{x^2}\,dx.

Example 6

easy
Does ∫1∞1x dx\int_1^{\infty}\frac{1}{x}\,dx converge or diverge?

Example 7

easy
For the pp-integral ∫1∞1xp dx\int_1^{\infty}\frac{1}{x^p}\,dx, when does it converge?

Example 8

easy
For ∫011xp dx\int_0^1\frac{1}{x^p}\,dx, when does it converge?

Example 9

easy
Rewrite ∫a∞f(x) dx\int_a^{\infty}f(x)\,dx using a limit.

Example 10

easy
Evaluate ∫011x dx\int_0^1\frac{1}{\sqrt{x}}\,dx.

Example 11

medium
Evaluate ∫0∞e−x dx\int_0^{\infty}e^{-x}\,dx.

Example 12

medium
Evaluate ∫1∞1x3 dx\int_1^{\infty}\frac{1}{x^3}\,dx.

Example 13

medium
Evaluate ∫0∞11+x2 dx\int_0^{\infty}\frac{1}{1+x^2}\,dx.

Example 14

medium
Use comparison to decide if ∫1∞1x2+1 dx\int_1^{\infty}\frac{1}{x^2+1}\,dx converges.

Example 15

medium
Use comparison to decide if ∫1∞1x+1 dx\int_1^{\infty}\frac{1}{\sqrt{x}+1}\,dx converges.

Example 16

medium
Evaluate ∫01ln⁡x dx\int_0^1\ln x\,dx.

Example 17

medium
Evaluate ∫2∞1xln⁡x dx\int_2^{\infty}\frac{1}{x\ln x}\,dx.

Example 18

medium
Evaluate ∫−∞∞11+x2 dx\int_{-\infty}^{\infty}\frac{1}{1+x^2}\,dx.

Example 19

medium
Evaluate ∫0∞e−2x dx\int_0^{\infty}e^{-2x}\,dx.

Example 20

challenge
Evaluate ∫0∞xe−x dx\int_0^{\infty}xe^{-x}\,dx.

Example 21

challenge
Find all pp for which ∫0∞1xp dx\int_0^{\infty}\frac{1}{x^p}\,dx converges.

Example 22

challenge
Evaluate ∫1∞ln⁡xx2 dx\int_1^{\infty}\frac{\ln x}{x^2}\,dx.

Example 23

easy
Does ∫1∞1x dx\int_1^{\infty}\frac{1}{\sqrt{x}}\,dx converge or diverge?

Example 24

easy
Which is the type of improperness in ∫3∞1x2+1 dx\int_3^{\infty}\frac{1}{x^2+1}\,dx?

Example 25

easy
Which is the type of improperness in ∫041x dx\int_0^4\frac{1}{\sqrt{x}}\,dx?

Example 26

easy
Does ∫011x2 dx\int_0^1\frac{1}{x^2}\,dx converge or diverge?

Example 27

medium
Evaluate ∫−∞0ex dx\int_{-\infty}^0 e^x\,dx.

Example 28

medium
Use the pp-test to classify ∫011x2/3 dx\int_0^1\frac{1}{x^{2/3}}\,dx.

Example 29

medium
Use comparison to decide whether ∫1∞sin⁡2xx2 dx\int_1^{\infty}\frac{\sin^2 x}{x^2}\,dx converges.

Example 30

medium
Evaluate ∫0∞2(x+1)3 dx\int_0^{\infty}\frac{2}{(x+1)^3}\,dx.

Example 31

medium
Evaluate ∫011x3 dx\int_0^1\frac{1}{\sqrt[3]{x}}\,dx.

Example 32

medium
Evaluate ∫1∞1x(x+1) dx\int_1^{\infty}\frac{1}{x(x+1)}\,dx.

Example 33

hard
Evaluate ∫0∞x(1+x2)2 dx\int_0^{\infty}\frac{x}{(1+x^2)^2}\,dx.

Example 34

hard
Use comparison to show ∫1∞1x2+x dx\int_1^{\infty}\frac{1}{x^2+\sqrt{x}}\,dx converges.

Example 35

hard
Evaluate ∫0∞1ex+e−x dx\int_0^{\infty}\frac{1}{e^x+e^{-x}}\,dx.

Example 36

challenge
Evaluate ∫0∞arctan⁡x1+x2 dx\int_0^{\infty}\frac{\arctan x}{1+x^2}\,dx.
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Background Knowledge

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

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