Question

Integrate:

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

Answer 1

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

let $x-1=m$ or $m+1=x$

$dx=dm$

$\Rightarrow \int \frac{(m-2)e^{m+1}}{m^3}dm$

Here (1) $u=(m-2)e^{m+1},V=\frac{1}{m^3}$

$(\because \int uvdx=u\int vdx-\int [\frac{dv}{dx}\int vdx])$

$\Rightarrow \frac{-e^{m+1}(m-2)}{2m^2}-\int -\frac{e^{m+1}+e^{m+1}m}{2m^2}dm$

$-\frac{1}{2}\int \frac{-e^{m+1}+e^{m+1}m}{m^2}dm$

Here (2)$m=e^{m+1}+e^{m+1}m,v=\frac{1}{m^2}$

$\Rightarrow \frac{-1}{2}[-\left(\frac{-e^{m+1}+e^{m+1}m}{m}\right)-e^{m+1}dn]$

$(-e^{m+1})$

$\Rightarrow \frac{-1}{2}(-\frac{e^{m+1}+e^{m+1}m}{m}-(-e^{1+m}\left)\right)$

$\frac{-e^{m+1}}{2m}$

$\Rightarrow \frac{-e^{m+1}(m-2)}{2m^2}+\frac{e^{m+1}}{2m}+c$

or

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