- #1
elimenohpee
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Homework Statement
Use the residue theorem to evaluate: [tex]\int^{\pi}_{0}\frac{16cos^{2}xdx}{5-4cosx}[/tex]
The Attempt at a Solution
I rewrote the integral with the substitutions
[tex] z=e^{ix} [/tex]
[tex] dx = \frac{dz}{iz} [/tex]
[tex] cosx = 0.5(z+z^{-1}) [/tex]
[tex] cos^{2}x = 0.25(z+z^{-1})^{2} [/tex]
I throw all that in, convert the integral then evaluate the poles. There are two simple poles at z=2 and z=0.5. I discard the z=2 pole since it lies outside the contour. I calculate the residue at z=0.5
[tex]\int_{C}\frac{4(z+z^{-1})^{2}dz}{-i(z-2)(2z-1)}[/tex]
[tex] res_{z->0.5}= ... = 25/3i [/tex]
therefore, it makes the integral = [tex] \frac{50\pi}{3} [/tex]
when I type this into wolfram alpha, I get 10 pi/ 3 ... any idea where I messed up?
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