Question

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

Answer 1

$$\begin{gathered} \mathrm{Let}I=\int \frac{e^x}{x}\left[x{\left(\log x\right)}^2+2\log x\right]dx \\ =\int e^x\left[{\left(\log x\right)}^2+\frac{2\log x}{x}\right]dx \\ \mathrm{Here},f\left(x\right)={\left(\log x\right)}^2 \\ \Rightarrow f\text{'}\left(x\right)=\frac{2\log x}{x} \\ \mathrm{put}{e}^{x}f\left(x\right)=t \\ \Rightarrow e^x{\left(\log x\right)}^2=t \\ \mathrm{Diff}\mathrm{both}\mathrm{sides}\mathrm{w}.\mathrm{r}.\mathrm{t}x \\ \left[e^x{\left(\log x\right)}^2+e^x\frac{2\log x}{x}\right]dx=dt \\ \therefore I=\int dt \\ =t+C \\ =e^x{\left(\log x\right)}^2+C \end{gathered}$$