# Maths - Integration by Substitution

> Source: https://ollybritton.com/notes/a-level/maths/topics/integration-by-substiution/ · Updated: 2021-03-29 · Tags: maths, further-maths, integration, school

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

## Flashcards
##### $$\int (x+2)^5 dx$$ What subsitution could you make in terms of $u$??
$$
u = x + 2
$$

##### If $u = x + 2$, what is $\frac{du}{dx}$??
$$
\frac{du}{dx} = 1
$$

##### If $u = x + 2$ and $\frac{du}{dx} = 1$, what is $dx$??
$$
du
$$

##### $$\int (x+2)^5 dx$$ If $u = x+2$ and $du = dx$, how could you rewrite the integral??
$$
\int u^5 du
$$

##### $$\int u^5 du$$ What is this equal to??
$$
\frac{u^6}{6} + c
$$

##### $$\frac{u^6}{6} + c$$ Can you reverse the subsitution $u = x + 2$??
$$
\frac{(x + 2)^6}{6} + c
$$

### 2021-03-30
##### If the subsitution for an integral is $u = x^2$ and the bounds are $[2, 4]$, what are the new bounds??
$$
[4, 16]
$$

##### $$\int \sqrt{1 + x^2}$$ What subsitution would you make??
$$
u = \sinh x
$$

##### $$\int \sqrt{1 - x^2}$$ What subsitution would you make??
$$
u = \sin x
$$

or

$$
u = \cos x
$$

##### $$\int \sqrt{x^2 - 1}$$ What subsitution would you make??
$$
u = \cosh x
$$

##### $$\int \sqrt{4 + 4^2}$$ What subsitution would you make??
$$
u = 2\sinh x
$$

### 2022-04-12
##### $$\int \frac{1}{2} x^3 \sqrt[3]{4 - x^2} \text{d}x$$ What substitution should you make here??
$$
u = 4 - x^2
$$

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