$\cfrac{{\cos 2x - \cos 2\alpha }}{{\cos x - \cos \alpha }}$

: Let $I = \int {\cfrac{{\cos 2x - \cos 2\alpha }}{{\cos x - \cos \alpha }}dx}$

$= \int {\cfrac{{\left( {2{{\cos }^2}x - 1} \right) - \left( {2{{\cos }^2}\alpha - 1} \right)}}{{\cos x - \cos \alpha }}} dx$

$= \int {\cfrac{{2\left( {{{\cos }^2}x - {{\cos }^2}\alpha } \right)}}{{\cos x - \cos \alpha }}} dx = \int {\cfrac{{2\left( {\cos x - \cos \alpha } \right)\left( {\cos x + \cos \alpha } \right)}}{{\cos x - \cos \alpha }}} dx$

$= 2\int {\left( {\cos \,x + \cos \alpha } \right)dx} = 2\int {\cos \,xdx} + 2\cos \alpha \int {dx}$

$= 2\sin x + 2x\cos \alpha + C = 2\left( {\sin x + x\cos \alpha } \right) + C$

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