# Further Maths - Improper Integrals

> Source: https://ollybritton.com/notes/a-level/further-maths/topics/improper-integrals/ · Updated: 2021-04-27 · Tags: further-maths, school, methods-in-calculus

## See Also
- [Maths - Integration](https://ollybritton.com/notes/a-level/maths/topics/integration/)
- [Maths - Integration by Substitution](https://ollybritton.com/notes/a-level/maths/topics/integration-by-substiution/)

## Flashcards
##### What two things mean an integral is improper??
* One or both of its limits are infinite
* It is undefined anywhere in $[a, b]$.

##### What do you call an improper integral that exists (has a defined value)??
Convergent.

##### What do you call an improper integral that does not exist??
Divergent.

##### $$\int^\infty_0 e^{-x} dx$$ What would you write instead to see if it has a defined value??
$$
\lim_{t \to \infty} \int^t_0 e^{-x}dx
$$

### 2022-05-12
##### $$\int^{5}_{0} \frac{2}{(2-x)^\frac{1}{3}} \text{d}x$$ How would you write this with limits as it is an improper integral??
$$
\lim_{\alpha \to 2^{-}} \int^{\alpha}_{0} \frac{2}{(2-x)^\frac{1}{3}} \text{d} x

+

\lim_{\beta \to 2^{+}} \int^{5}_{\beta} \frac{2}{(2-x)^\frac{1}{3}} \text{d} x
$$

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