Practice Improper Integrals in Math

Use these practice problems to test your method after reviewing the concept explanation and worked examples.

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Quick 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.

Showing a random 20 of 50 problems.

Example 1

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

Example 2

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

Example 3

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

Example 4

easy
Evaluate ∫1∞1x4 dx\int_1^{\infty}\frac{1}{x^4}\,dx.

Example 5

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

Example 6

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

Example 7

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

Example 8

medium
Evaluate ∫−∞∞xe−x2 dx\int_{-\infty}^{\infty}xe^{-x^2}\,dx.

Example 9

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

Example 10

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

Example 11

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

Example 12

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

Example 13

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

Example 14

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

Example 15

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

Example 16

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

Example 17

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

Example 18

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

Example 19

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

Example 20

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