# Further Maths - Improper Integrals

## See Also

## 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?

\[\lim_{\alpha \to 2^{-}} \int^{\alpha}_{0} \frac{2}{(2-x)^\frac{1}{3}} \text{d} x
+
\lim_{\beta \to 2^{+}} \int^{5}_{\beta} \frac{2}{(2-x)^\frac{1}{3}} \text{d} x\]