Question

Integrate

$\int \frac{1}{\sin (x-a)\cos (x-b)}dx$

Answer 1

$\int \frac{dx}{\sin (x-a)\cos (x-b)}$

$\Rightarrow \frac{1}{\cos (b-a)}\int \frac{\cos (b-a)dx}{\sin (x-a)\cos (x-b)}$

$\Rightarrow \frac{1}{\cos (b-a)}\int \frac{\cos (x-a)-(x-6)}{\sin (x-a)\cos (x-6)}dx$

$\Rightarrow \frac{1}{\cos (b-a)}\int \frac{\cos (x-a)\cos (x-6)+\sin (x-a)\sin (x-6)}{\sin (x-a)\cos (x-6)}dx$

$\Rightarrow \frac{1}{cos(b-a)}\int [cos(x-a)+\tan (x-6)]dx$

$\Rightarrow \sec (b-a)[In\left|\sec (x-6)\right|+In\left|\sin (x-a)\right|]+c$