Mathematics

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


ANSWER

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


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
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