QPaperGen

$$\cos 2x - \cos 2\alpha = (2\cos^2 x - 1) - (2\cos^2 \alpha - 1)$$$$\cos 2x - \cos 2\alpha = 2\cos^2 x - 2\cos^2 \alpha$$$$\cos 2x - \cos 2\alpha = 2(\cos^2 x - \cos^2 \alpha)$$
$$=2(\cos x - \cos \alpha)(\cos x + \cos \alpha)$$
Hence
$$\frac{\cos 2x-\cos 2\alpha}{\cos x-\cos \alpha} = \frac{2(\cos x - \cos \alpha)(\cos x + \cos \alpha)}{\cos x-\cos \alpha}$$
$$ = 2(\cos x + \cos \alpha)$$
Hence integral becomes:$$I = \int 2(\cos x + \cos \alpha) dx$$
$$I = \mathbf{2(\sin x + x \cos \alpha) + C}$$