Question

$\int \frac{e^x\left(x-4\right)}{{\left(x-2\right)}^3}dx$

Answer 1

$$\begin{gathered} \mathrm{Let}I=\int e^x\left(\frac{x-4}{{\left(x-2\right)}^3}\right)dx \\ =\int e^x\left[\frac{x-2-2}{{\left(x-2\right)}^3}\right]dx \\ =\int e^x\left[\frac{1}{{\left(x-2\right)}^2}-\frac{2}{{\left(x-2\right)}^3}\right]dx \\ \mathrm{Here},f\left(x\right)=\frac{1}{{\left(x-2\right)}^2} \\ \Rightarrow f\text{'}\left(x\right)=\frac{-2}{{\left(x-2\right)}^3} \\ \mathrm{Put}{e}^{x}f\left(x\right)=t \\ \Rightarrow e^x\frac{1}{{\left(x-2\right)}^2}=t \\ \mathrm{Diff}\mathrm{both}\mathrm{sides}\mathrm{w}.\mathrm{r}.\mathrm{t}x \\ \left[e^x\frac{1}{{\left(x-2\right)}^2}+e^x\frac{-2}{{\left(x-2\right)}^3}\right]dx=dt \\ \therefore I=\int dt \\ =t+C \\ =\frac{e^x}{{\left(x-2\right)}^2}+C \end{gathered}$$