Question

Write a value of $\int \frac{1+\cot x}{x+\log \sin x}dx$.

Answer 1

$$\begin{gathered} \mathrm{Let}I=\int \frac{1+\cot x}{x+\log \sin x}dx \\ \mathrm{Let}x+\log \sin x=t \\ \Rightarrow \left(1+\frac{1}{\sin x}\times \cos x\right)dx=dt \\ \Rightarrow \left(1+\cot x\right)dx=dt \\ \therefore I=\int \frac{dt}{t} \\ =\log \left|t\right|+C \\ =\log \left|x+\log \sin x\right|+C \end{gathered}$$