I've got a couple of these sort of questions wrong, and its pretty demoralising. Is it the method I'm using, or is it just arithmetic/silly errors? Here's one:
Use trigonometric identities before using a substitution (or reversing the chain rule) to integrate the following.
part c) $\int \sin\left(x\right)\cos\left(x\right)e^{\cos2x}$
Sub $u=\cos(x)$
$\int \sin\left(x\right)ue^{\cos\left(2x\right)}dx=\int \sin\left(x\right)ue^{2u^2-1}dx\:$
$=-1\int ue^{2u^2-1}du\:$
Then integration by parts:
$\frac{u}{2}e^{2u-1}-\int \:\frac{1}{2}e^{2u^2-1}=\frac{u}{2}e^{2u-1}-\frac{1}{4}e^{2u^2-1}$
$=\frac{\cos\left(x\right)}{2}e^{2\cos^2x-1}-\frac{1}{4}e^{2\cos^2x-1}$
$-1\cdot \frac{\cos\left(x\right)}{2}e^{2\cos^2x-1}-\frac{1}{4}e^{2\cos^2x-1}=\frac{-\cos\left(x\right)}{2}e^{2\cos^2x-1}+\frac{1}{4}e^{2\cos^2x-1}$