Practice Improper Integrals in Math

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

Worksheet PDF Free Answer-key PDF Family

Quick Recap

Integrals where the interval of integration is infinite (Type I: ∫a∞f(x) dx) or the integrand has an infinite discontinuity on the interval (Type II: ∫abf(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 1x2 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 p-test to classify ∫011x2/3 dx.

Example 2

easy
Rewrite ∫a∞f(x) dx using a limit.

Example 3

challenge
Evaluate ∫0∞arctan⁡x1+x2 dx.

Example 4

easy
Evaluate ∫1∞1x4 dx.

Example 5

easy
Evaluate ∫011x dx.

Example 6

medium
Evaluate ∫011x3 dx.

Example 7

hard
Evaluate ∫1∞1x2(x+1) dx.

Example 8

medium
Evaluate ∫−∞∞xe−x2 dx.

Example 9

easy
For the p-integral ∫1∞1xp dx, when does it converge?

Example 10

easy
Does ∫1∞1x dx converge or diverge?

Example 11

medium
Evaluate ∫0∞11+x2 dx.

Example 12

medium
Evaluate ∫−∞∞11+x2 dx.

Example 13

easy
Does ∫011x2 dx converge or diverge?

Example 14

easy
For ∫011xp dx, when does it converge?

Example 15

hard
Evaluate ∫0∞x(1+x2)2 dx.

Example 16

medium
Evaluate ∫0∞2(x+1)3 dx.

Example 17

medium
Use comparison to decide if ∫1∞1x+1 dx converges.

Example 18

challenge
Find all p for which ∫0∞1xp dx converges.

Example 19

easy
Evaluate ∫1∞1x2 dx.

Example 20

medium
Evaluate ∫0∞xe−2x dx.
← Back to Improper Integrals Worked Examples Formula Explained