Mathematics

Evaluate $\displaystyle \int\frac{x+9}{(x+10)^{2}}e^{x}dx=$

$e^{x}\dfrac{1}{x+10}+c$

SOLUTION
$\int e^{x}\dfrac{x+9}{(x+10)^{2}}dx$
$=\int e^{x}\dfrac{[(x+10)-1]}{(x+10)^{2}}dx$
$=\int e^{x}\left [ \dfrac{1}{(x+10)}-\dfrac{1}{(x+10)^{2}} \right ]dx$
It is in the form of
$\int e^{x}(f(x)+f^{'}(x))dx$
$=e^{x}f(x)+c$
$\Rightarrow \ \int e^{x}\left [ \dfrac{1}{(x+10)}-\dfrac{1}{(x+10)^{2}} \right ]dx$
$=\dfrac{e^{x}}{(x+10)}+c$
$\therefore \displaystyle \int \dfrac{x+9}{(x+10)^{2}}e^{x}\ dx=\dfrac{e^{x}}{(x+10)}+c$

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Single Correct Medium Published on 17th 09, 2020
Questions 203525
Subjects 9
Chapters 126
Enrolled Students 109

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