First slide

Methods of integration

Question

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

Moderate

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 $f (x) = \frac{1}{(x - 1)^{2}} \Rightarrow f^{'} (x) = \frac{- 2}{(x - 1)^{3}}$
$∴ I = \frac{e^{x}}{(x - 1)^{2}} + C [∵ \int e^{x} \{f (x) + f^{'} (x)\} dx = e^{x} f (x)]$

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