Question

Find $\int \frac{e^x(x-3)}{{(x-1)}^3}dx$

Answer 1

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

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

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

$=\frac{1}{{(x-1)}^2}\int e^{x}dx-\int \left[\frac{d}{dx}\left\{\frac{1}{{(x-1)}^2}\right\}{e}^{x}dx\right]-2\int \frac{e^x}{{(x-1)}^3}dx+c$

$=\frac{e^x}{{(x-1)}^2}+2\int \frac{e^x}{{(x-1)}^3}dx-2\int \frac{e^x}{{(x-1)}^3}dx+c$

$=\frac{e^x}{{(x-1)}^2}+c$