Complex Integral I just can't figure out

[~Death~]

hi, this is really frustrating because I've been working on this one integral for the last 2 1/2 horus and i can't figure out what I did wrong...

it's the Integral sin^2(x)/(3-2cos(x)) dx from x=0 to x=2pi


I tried the substitution z=e^(itheta) plugging in sin and cos as functions of z and I

ultimately get:

i * integral (z^4+2z^2+1)/(z^2-3z+1)dz over |z|=1

Then the poles inside the unit circle are @ z=0 and (3-sqrt(5))/2

so I find the residue @ z=0 to be 3 and @ (3-sqrt5)/2 something like

(looking in my mess of notes) 4(15-6sqrt(5))^2/((3-sqrt(5)^2(-sqrt(5))

then I plug in 2pi X Sum residues X i

but it doesn't agree with the answer I get on my calculator which is around 1.19

any help is really appreciated thanks!

 

[~Death~]

sorry to anyone who was trying to figure it out...i solved it

turns out i substitutes 1/2(z+1/z) for sine instead of 1/(2i)(z-1/z)

I make the stupidest mistakes ever and it costs me. anyone know how i can

avoid these =(