Question

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

Answer 1

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

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

Using $\int u.vdx=u\int vdx$ $-\int (\frac{du}{dx}\int vdx)dx$

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

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

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