Mathematics

Integrate:
$$\int { \dfrac { \left( x-3 \right) { e }^{ x } }{ { \left( x-1 \right)  }^{ 3 } } dx }$$


SOLUTION
$$ \displaystyle \int \dfrac{(x-3)e^{x}}{(x-1)^{3}}dx $$ (By ILATE)
let $$x-1 = m$$ or $$m+1 = x $$
$$ dx =dm$$
$$ \Rightarrow  \displaystyle \int  \dfrac{(m-2)e^{m+1}}{m^{3}}dm$$
Here (1) $$ u = (m-2)e^{m+1}, V = \dfrac{1}{m^{3}}$$
$$ (\because \displaystyle \int uv \, dx = u \int v dx - \int \left [ \dfrac{dv}{dx} \int v \,dx \right ])$$
$$ \Rightarrow \dfrac{-e^{m+1}(m-2)}{2m^{2}} - \displaystyle \int -\frac{e^{m+1}+e^{m+1}m}{2m^{2}}dm $$
$$ \displaystyle  -\dfrac{1}{2} \int \dfrac{-e^{m+1}+e^{m+1}m}{m^{2}}dm $$
Here (2)$$ m = e^{m+1}+e^{m+1} m , v = \dfrac{1}{m^{2}}$$
$$ \Rightarrow \frac{-1}{2} \left [ -\left ( \dfrac{-e^{m+1}+e^{m+1}m}{m} \right )-e^{m+1} dn \right ]$$
$$ (-e^{m+1})$$
$$ \Rightarrow  \dfrac{-1}{2} \left ( -\dfrac{e^{m+1}+e^{m+1}m}{m} -(-e^{1+m})\right )$$
$$ \dfrac{-e^{m+1}}{2m}$$
$$ \Rightarrow \dfrac{-e^{m+1}(m-2)}{2m^{2}} + \dfrac{e^{m+1}}{2m}+c $$
or
$$ \Rightarrow  \boxed{ \dfrac{e^{x}(x-3)}{2(x-1)^{2}}+ \dfrac{e^{x}}{2(x-1)}+c}$$
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Subjective Medium Published on 17th 09, 2020
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