Question

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

Answer 1

$\begin{array}{c}I=\int \frac{1+x}{x\sin \left(x+\log x\right)}dx\\ =\int \frac{\left(\frac{1+x}{x}\right)}{\sin \left(x+\log x\right)}dx\\ Let,x+\log x=t\\ \left(1+\frac{1}{x}\right)dx=dt\\ \left(\frac{x+1}{x}\right)dx=dt\\ I=\int \frac{dt}{\sin \left(t\right)}=\int \mathrm{cosec}tdt\\ =-\ln |\mathrm{cosec}t+\cot t|+C\\ =-\ln |\mathrm{cosec}\left(x+\log x\right)+\cot \left(x+\log x\right)|+C\end{array}$