Hey guys.
Any idea about how to solve this integral?
10x.
Hello, this can be solved analytically letting $\displaystyle \theta=2\arctan(\phi)$
NOTE Take what I am about to say with extreme caution...I am a novice at this and just learning so wait for confirmation!
I think this integral can also be solved using Residues by considering the function
$\displaystyle f(z)=\frac{i}{(z-a)(az-1)}$ and taking
$\displaystyle \int_{|z|=1}f(z)~dz$
Substitute $\displaystyle z = e^{it}$:
1. $\displaystyle dt = \frac{dz}{iz} = \frac{-i \, dz}{z}$.
2. $\displaystyle \cos t = \frac{e^{it} + e^{-it}}{2} = \frac{z + \frac{1}{z}}{2} = \frac{z^2 + 1}{2z}$.
Simplify. The integrand becomes $\displaystyle \frac{i}{az^2 - z - a^2 z + a}$ which factorises in the way Mathstud gave.