Practice Improper Integrals in Math
Use these practice problems to test your method after reviewing the concept explanation and worked examples.
Quick 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.
Example 1
easyEvaluate \displaystyle\int_1^{\infty} \frac{1}{x^2}\,dx.
Example 2
hardEvaluate \displaystyle\int_0^1 \frac{1}{\sqrt{x}}\,dx (Type II).
Example 3
easyDoes \displaystyle\int_1^{\infty} \frac{1}{x}\,dx converge or diverge?
Example 4
mediumEvaluate \displaystyle\int_0^{\infty} e^{-x}\,dx.