Question

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

Answer 1

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

$I=\int e^x[\left(\frac{x^2-1}{(x-1{)}^2}\right)-\left(\frac{2}{(x-1{)}^2}\right)]dx$

$I=\int e^x[\frac{x+1}{x-1}+\frac{(-2)}{(x-1{)}^2}]dx$

$\frac{d}{dx}\left(\frac{x+1}{x-1}\right)=\frac{-2}{(x-1{)}^2}$

$\therefore$ It is of the form $\int e^x\left(f\right(x)+f^{\prime }(x\left)\right)dx$

$\therefore I=e^x\left(\frac{x+1}{x-1}\right)$