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: \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.

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: Replace the 'infinity' or 'blow-up point' with a variable, compute the integral, then take the limit. If the limit is finite, the integral converges; if not, it diverges.

Common stuck point: For Type II, the discontinuity might be inside the interval, not at an endpoint. In that case, split the integral at the discontinuity and evaluate each piece as a separate limit.

Sense of Study hint: Replace the infinity or blow-up point with a variable like b, compute the integral normally, then take the limit as b approaches infinity (or the trouble spot).

Worked Examples

Example 1

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

Solution

  1. 1
    Replace the infinite upper limit with a variable: \int_1^{\infty}\frac{1}{x^2}\,dx = \lim_{b\to\infty}\int_1^b x^{-2}\,dx
  2. 2
    Integrate x^{-2}: = \lim_{b\to\infty}\left[-\frac{1}{x}\right]_1^b = \lim_{b\to\infty}\left(-\frac{1}{b} + 1\right)
  3. 3
    Take the limit as b \to \infty: since \frac{1}{b} \to 0, the integral converges to 1.

Answer

1
Replace \infty with b, integrate, take the limit. The 1/b term vanishes.

Example 2

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

Practice Problems

Try these problems on your own first, then open the solution to compare your method.

Example 1

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

Example 2

medium
Evaluate \displaystyle\int_0^{\infty} e^{-x}\,dx.
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Background Knowledge

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

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