Question

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

Answer 1

We have,

$I=\int \frac{\cos \left(x+a\right)}{\cos \left(x-a\right)}dx$

$Putx-\alpha =tdx=dtx=\left(\alpha +t\right)$

Therefore,

$I=\int \frac{\cos \left(2\alpha +t\right)}{\cos t}dx=\int \frac{\cos 2\alpha .\cos t-\sin 2\alpha .\sin t}{\cos t}dt=\int \left(\cos 2\alpha -\sin 2\alpha .\tan t\right)dt=t.\cos 2\alpha +\sin 2\alpha .\log \left|\cos t\right|+C$

On putting the value of $t$, we get

$I=\left(x-\alpha \right)\cos 2\alpha +\sin 2\alpha .\log \left|\cos \left(x-\alpha \right)\right|+C$

Hence, this is the answer.