Question

$\int \left(\frac{x+1}{x}\right){\left(x+\log x\right)}^{2}dx$

Answer 1

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