Question

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

Answer 1

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

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

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

$=\cos \left(a-b\right)\int \cot \left(x+b\right)dx-\sin \left(a-b\right)\int dx$

$=\cos \left(a-b\right)\log \left|\sin \left(x+b\right)\right|-x\sin \left(a-b\right)+C$