Question

Evaluate: $\int \frac{\cos 2x-\cos 2a}{\cos x-\cos a}dx$.

Answer 1

$\int \frac{\cos 2x-\cos 2a}{\cos x-\cos a}dx$

We know that $\cos 2x=2{\cos }^{2}x-1$

$=\int \frac{2{\cos }^{2}x-1-2{\cos }^{2}a+1}{\cos x-\cos a}dx$

$=2\int \frac{{\cos }^{2}x-{\cos }^{2}a}{\cos x-\cos a}dx$

$=2\int \frac{(\cos x-\cos a)(\cos x+\cos a)}{\cos x-\cos a}dx$

$=2\int (\cos x+\cos a)dx$

$=2(\sin x+x\cos a)+C$