1. ## Evaluate integral?

Evaluate:
$\int_{0}^{\pi} e^{|\cos x|} \left\{2\sin\left(\frac{1}{2}\cos x\right) + 3\cos\left(\frac{1}{2}\cos x\right)\right\}\sin x dx$

2. I think you want to break the integral into two parts by the sum. Then make the u substitution u=1/2cos(x). That will get you the sin(x)dx you need. so you will have a $-2e^{|2u|}(a)sin/cos(u)du$
May need to break this integral up again to deal with that absolute value.

Then I think just do the integration by parts twice trick and you will get back something familar and solve that way. I assume you know what you are doing if you were assigned this kind of problem, so I left out a lot of steps. Let me know if you need me to fill in details.

3. ooh, i just started working it out a bit and I got the first one to reduce to

$4\int_{-1/2}^{1/2}e^{|2u|}sin(u)du$

But this is an odd function, with symmetric limits of integration so it should be 0.

For the second one
I get $6\int_{-1/2}^{1/2}e^{|2u|}cos(u)du$ which is an even function, so this is going to be
$12\int_{0}^{1/2}e^{2u}cos(u)du$

which yeah should be able to just do the integration by parts twice to solve.

Be advised I did these on the back of an envelope really quicky, but hope that helps.