Fluency exercise

## Problem

Below are 8 integrals and their solutions. Can you:

• match up the cards and fill in the blanks?

• write down a substitution that could have been used to solve each integral?

• sort the pairs into groups. What features could be used to define the groups?

$\displaystyle{\int 9x\sqrt{1-9x^2} \, dx}$

$\displaystyle{\int \dfrac{x^2-1}{(x^3 - 3x)^2} \, dx}$

$\displaystyle{\int \dfrac{1}{\sqrt{1-9x^2}} \, dx}$

$\displaystyle{\int \dfrac{9x}{\sqrt{1-9x^2}} \, dx}$

$\displaystyle{\int \dfrac{6-6x^2}{x^3 - 3x} \, dx}$

$\displaystyle{\int \dfrac{\cos x}{\sin^3 x} \, dx}$

$\displaystyle{\int \dfrac{\cos x}{\sin x} \, dx}$

$\dfrac{-1}{2\sin^2x}+c$

$\frac{2}{9} (x^3 - 3x)^{3/2} +c$

$\ln{|\sin x|}+c$

$\frac{1}{3}\arcsin 3x +c$

$\dfrac{-1}{3(x^3-3x)}+c$

$-\frac{1}{3}\left(1-9x^2\right)^{\frac{3}{2}} +c$

$-2\ln{|x^3-3x|}+c$