# Maths - Integration by Substitution

## See Also

## 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?

\[u = 4 - x^2\]