Question

Solve $\int \frac{\left(1+\cot x\right)}{\left(x+\log \sin x\right)}dx$

Answer 1

$\int \frac{1+\cot x}{x+\log \left(\sin x\right)}dx$

$x+\log \left(\sin x\right)=t$

$\left(1+\frac{\cos x}{\sin x}\right)dx=dt$

$(1+\cot x)dx=dt$

$\int \frac{dt}{t}$

$=\log t+c$

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