Question

$\int \frac{1+\cos x}{{\left(x+\sin x\right)}^3}dx$

Answer 1

$$\begin{gathered} \int \frac{\left(1+\cos x\right)}{{\left(x+\sin x\right)}^3}dx \\ \mathrm{Let}x+\sin x=t \\ \Rightarrow \left(1+\cos x\right)=\frac{dt}{dx} \\ \Rightarrow \left(1+\cos x\right)dx=dt \\ \mathrm{Now},\int \frac{\left(1+\cos x\right)}{{\left(x+\sin x\right)}^3}dx \\ =\int \frac{dt}{t^3} \\ =\int t^{-3}dt \\ =\frac{t^{-3+1}}{-3+1}+\mathrm{C} \\ =\frac{-1}{2t^2}+\mathrm{C} \\ =\frac{-1}{2{\left(x+\sin x\right)}^2}+\mathrm{C} \end{gathered}$$