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?

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

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

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

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

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

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

\(\displaystyle{\int \dfrac{\cos x}{\sin x} \, dx}\)

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

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

\(\ln{|\sin x|}+c\)

\(\frac{1}{3}\arcsin 3x +c\)

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

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

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