AcademyIntegral Calculus
Academy
Improper Integrals
Level 1 - Calculus topic page in Integral Calculus.
Principle
An improper integral has an infinite interval, an unbounded integrand, or both. It is defined using limits, and it converges only if those limits exist as finite numbers.
Notation
\(\int_a^\infty f(x)\,dx\)
improper integral over an infinite interval
\(\lim_{b\to\infty}\)
limit used to define an infinite endpoint
\(p\)
power in a p-integral
The Core Method
Replace the improper endpoint or singular point by a finite variable, integrate, and take the limit.
Infinite interval
\[\int_a^\infty f(x)\,dx=\lim_{b\to\infty}\int_a^b f(x)\,dx\]
For \(p\)-integrals, \(\int_1^\infty x^{-p}dx\) converges exactly when \(p>1\).
Worked Cases
Question
Evaluate \(\int_1^\infty \frac{1}{x^2}\,dx\).
Answer
Write the integral as \(\lim_{b\to\infty}\int_1^b x^{-2}dx\). An antiderivative is \(-x^{-1}\). Evaluate: \([-1/x]_1^b=-1/b+1\). Taking \(b\to\infty\) gives \(1\).
Examples
Question
Does \(\int_1^\infty \frac{1}{x}\,dx\) converge?
Answer
Use a limit: \(\int_1^b \frac1x dx=\ln b-\ln1=\ln b\). As \(b\to\infty\), \(\ln b\to\infty\). Therefore the improper integral diverges.