Mathematics

Evaluate the following: $\displaystyle\int \dfrac { 2 x + 7 } { ( x - 4 ) ^ { 2 } } d x$

SOLUTION
$I=\displaystyle\int\dfrac{2x+7}{(x-4)^2}dx$

$I=\displaystyle\int\dfrac{2x-8+15}{(x-4)^2}dx=\displaystyle\int\dfrac{2x-8}{(x-4)^2}dx+\displaystyle\int\dfrac{15}{(x-4)^2}dx$

$I=\displaystyle\int\dfrac{2(x-4)}{(x-4)^2}dx+15\displaystyle\int\dfrac{1}{(x-4)^2}dx$

Let $x-4=t\Rightarrow dx=dt$

$I=\displaystyle2\int \dfrac{dx}{t}+15\int \dfrac{1}{t^2}dt$

$I=2\log(t)+15\dfrac{t^{-1}}{-1}$

$I=2log(t)-\dfrac{15}{t}$

$I=2log (x-4)-\dfrac{15}{(x-4)}+c$.

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Subjective Medium Published on 17th 09, 2020
Questions 203525
Subjects 9
Chapters 126
Enrolled Students 86

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