# Improper integral

An integral with an infinite limit, or a function that blows up inside the interval. Replace the problem point with a variable and take a limit.

Maths (university) · Part 06: Integration techniques and uses · Card 5 of 9 · Last updated 2026-10-19

## Analogy

A road that goes on forever: you ask whether the total distance walked settles to a finite number.

## Example

**Integral of 1/x^2 from 1 to infinity**

Integral from 1 to R is 1 - 1/R.
As R tends to infinity, 1/R tends to 0.
Limit: 1.
(1/x from 1 to infinity diverges.)

## In practice

Used in probability, physics and for deciding whether a series converges.

## Practice

### Flashcards

- **What is the integral of 1/x^2 from 1 to infinity?** 1.

### Interview questions

**Does the integral of 1/x^2 from 1 to infinity converge? Compare with 1/x.**

- Replace infinity with R and take a limit
- Integral of 1/x^2 from 1 to R is 1 - 1/R
- Limit is 1, so it converges
- Integral of 1/x gives ln R, which diverges

## Questions people ask

### How can an infinite area be finite?

The curve gets close to the x-axis fast enough that the thin tail adds up to a small amount. For 1 over x squared on 1 to infinity, the area is exactly 1. The slower 1 over x does not shrink fast enough, so its area is infinite.

### What does it mean for an improper integral to diverge?

The limit either does not exist or is infinite, so the area has no finite value. For 1 over x from 1 to R, the answer is ln R, which grows without bound as R grows. We then say the integral diverges.

## Related concepts

- [Integration by parts](https://studyvisually.com/learn/science/maths-university/maths-university-part-6/maths-university-part-6-concept-1)
- [Choosing u and dv](https://studyvisually.com/learn/science/maths-university/maths-university-part-6/maths-university-part-6-concept-2)
- [Partial fractions](https://studyvisually.com/learn/science/maths-university/maths-university-part-6/maths-university-part-6-concept-3)
- [Trigonometric integrals](https://studyvisually.com/learn/science/maths-university/maths-university-part-6/maths-university-part-6-concept-4)

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