Question

$\int e^x\left(\frac{1}{x^2}-\frac{2}{x^3}\right)dx$

Answer 1

$$\begin{gathered} \mathrm{Let}I=\int e^x\left(\frac{1}{x^2}-\frac{2}{x^3}\right)dx \\ \mathrm{Also}\mathrm{let}{e}^x\times \frac{1}{x^2}=t \\ \mathrm{Diff}\mathrm{both}\mathrm{sides}\mathrm{w}.\mathrm{r}.\mathrm{t}x \\ {e}^x\times \frac{1}{x^2}+e^x\left(\frac{-2}{x^3}\right)=\frac{dt}{dx} \\ \Rightarrow e^x\left(\frac{1}{x^2}-\frac{2}{x^3}\right)dx=dt \\ \therefore \int e^x\left(\frac{1}{x^2}-\frac{2}{x^3}\right)dx=\int dt \\ =t+C \\ =\frac{e^x}{x^2}+C \end{gathered}$$