Question

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

Answer 1

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

$=\int \frac{\cos \left(t-a\right)}{\sin t}dt$

$=\int \left(\frac{\cos t\cos a}{\sin t}+\frac{\sin t\sin a}{\sin t}\right)dt$

$=\cos a\int \cot t+\sin a\int dt$

$=\cos a\log \left|\sin t\right|+\sin a\left(t\right)+c$

$=\cos a\log \left|\sin \left(x+a\right)\right|+\left(x+a\right)\sin a+c$