Question

Find the integral :
$\int \frac{\cos x}{\cos (x-a)}dx$

Answer 1

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

$\Rightarrow \int \frac{cos\left(\right(x-a)+a)}{cos(x-a)}dx=[\int \frac{cos(x-a)\times cosa}{cos(x-a)}+\int \frac{sin(x-a)\times sina}{cos(x-a)}]dx$

$=[\int cos(a)+\int tan(x-a)\times sin(a\left)\right]dx$

$=xcos\left(a\right)+sin\left(a\right)\times log\left(secx\right)+C$

$\int \frac{cos\left(x\right)}{cos(x-a)}dx=xcos\left(a\right)+sin\left(a\right)\times log\left(secx\right)+C$