Mathematics

Evaluate the following integral:$\displaystyle \int { \cfrac { 2x-1 }{ { \left( x-1 \right) }^{ 2 } } } dx\quad \quad$

SOLUTION
$\displaystyle\int{\dfrac{2x-1}{{\left(x-1\right)}^{2}}dx}$

$=\displaystyle\int{\dfrac{2x-2+1}{{\left(x-1\right)}^{2}}dx}$

$=\displaystyle\int{\dfrac{2\left(x-1\right)}{{\left(x-1\right)}^{2}}dx}+\displaystyle\int{\dfrac{dx}{{\left(x-1\right)}^{2}}}$

$=2\displaystyle\int{\dfrac{dx}{\left(x-1\right)}}+\displaystyle\int{\dfrac{dx}{{\left(x-1\right)}^{2}}}$

Let $(x-1)=t\ \Rightarrow\ dx=dt$

$=2\displaystyle\int{\dfrac{dx}{\left(t\right)}}+\displaystyle\int{\dfrac{dx}{{\left(t\right)}^{2}}}$

$=2\log{\left|t\right|}-\dfrac{1}{t}+c$           $\left[\because \displaystyle\int{\dfrac{dx}{{x}^{2}}}=\dfrac{-1}{x},\int{\dfrac{1}{x}}dx=\log |x|\right]$

$=2\log{\left|x-1\right|}-\dfrac{1}{x-1}+c$ where $t=x-1$

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Subjective Medium Published on 17th 09, 2020
Questions 203525
Subjects 9
Chapters 126
Enrolled Students 84

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