Question

Integrate the function $\frac{(x-3)e^x}{(x-1{)}^3}$

Answer 1

Let $I=\int e^x\left\{\frac{x-3}{(x-1{)}^3}\right\}dx=\int e^x\left\{\frac{x-1-2}{(x-1{)}^2}\right\}dx$
$=\int e^x\{\frac{1}{(x-1{)}^2}-\frac{2}{(x-1{)}^3}\}dx$
Let $f\left(x\right)=\frac{1}{(x-1{)}^2}\Rightarrow f^{\prime }\left(x\right)=\frac{-2}{(x-1{)}^3}$
Now know that, $\int e^x\left\{f\right(x)+f^{\prime }(x\left)\right\}dx=e^{x}f\left(x\right)+C$
$\therefore I=\frac{e^x}{(x-1{)}^2}+C$