Question

Integrate
$\int \cos 5x.\cos 3xdx$

Answer 1

Let,

$y=\int \cos 5\theta \cos 3\theta d\theta$ ……(1)

$y=\frac{1}{2}\int 2\cos 5\theta \cos 3\theta d\theta$

$\because 2\cos A\cos B=\cos (A+B)+\cos (A-B)$

Therefore,

$y=\int (\cos 8x+\cos 2x)d\theta$

$=\int \cos 8xdx+\int \cos 2xdx+C$

$=\frac{-\sin 2\theta }{8}+\frac{-\sin 2\theta }{2}+C$

Hence, this is the answer.