Question

Find the integral of the function
$\frac{\cos 2x-\cos 2\alpha }{\cos x-\cos \alpha }$ with respect to x

Answer 1

Consider the given function

$\int \frac{cos2x-cos2\alpha }{cosx-cos\alpha }dx$

$\Rightarrow \int \frac{2{\cos }^{2}x-1-2co{s}^2\alpha +1}{cosx-cos\alpha }dx\therefore \cos 2\theta =2{\cos }^2\theta -1$

$\Rightarrow \int \frac{2{\cos }^{2}x-2co{s}^2\alpha }{cosx-cos\alpha }dx$

$\Rightarrow 2\int \frac{{\cos }^{2}x-co{s}^2\alpha }{cosx-cos\alpha }dx$

$\Rightarrow 2\int \frac{(\cos x-cos\alpha )(\cos x+cos\alpha )}{(cosx-cos\alpha )}dx$

$\Rightarrow 2\int (\cos x+cos\alpha )dx$

$\Rightarrow 2\int \cos xdx+2\int \cos \alpha dx$

$\Rightarrow 2\sin x+2x\cos \alpha +C$

Hence, this is the answer.