∫x−3(x−1)3⋅exdx is equal to - Infinity Learn by Sri Chaitanya

A
$\frac{e^{x}}{(x - 1)} + c$
B
$\frac{- e^{x}}{(x - 1)^{2}} + C$
C
$\frac{2 e^{x}}{(1 - x)^{2}} + C$
D
$\frac{e^{x}}{(x - 1)^{2}} + C$

Solution:

Let $\begin{aligned} I = \int \frac{x - 3}{(x - 1)^{3}} e^{x} dx = \int \frac{x - 1 - 2}{(x - 1)^{3}} e^{x} dx \\ = \int e^{x} [\frac{x - 1}{(x - 1)^{3}} - \frac{2}{(x - 1)^{3}}] dx \end{aligned}$

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

Let $f (x) = \frac{1}{(x - 1)^{2}} \Rightarrow f^{'} (x) = \frac{- 2}{(x - 1)^{3}}$

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

$\int \frac{x - 3}{(x - 1)^{3}} \cdot e^{x} dx$ is equal to

Solution:

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