How to integrate $\cos^3x$ by parts I've converted $\cos^3(x)$ into $\cos^2(x)\cos(x)$ but still have not gotten the answer. 
The answer is $\dfrac{\sin(x)(3\cos^2x + 2\sin^2x)}{3}$
My answer was the same except I did not have a $3$ infront of $x$ and my $2\sin^2x$ was not squared.
Help! 

Since $\cos^3x=(1-\sin^2 x)\cos x$, you can do $\sin x=t$ and $\cos x\,\mathrm dx=\mathrm dt$, thereby getting$$\int1-t^2\,\mathrm dt.$$
