Mathematics

# $\displaystyle \int \frac{2x^{12}+ 5x^9}{(x^5+x^3+1)^3}dx$ is equal to

$\dfrac{x^{10}}{2(x^5 +x^3 +1)^2}+C$

##### SOLUTION
$\displaystyle \int \frac{2x^{12}+ 5x^9}{(x^5+x^3+1)^3}dx$

Divide numerator & denominator by highest power of expression i.e., '$x^{15}$'

$\displaystyle\int \dfrac{\dfrac{2}{x^3}+\dfrac{5}{x^6}}{\left(1+\dfrac{1}{x^2}+\dfrac{1}{x^5}\right)^3}dx$

Put $1+\dfrac{1}{x^2}+\dfrac{1}{x^5}=t$

$\Rightarrow \dfrac{-2}{x^3}-\dfrac{5}{x^6}dx=dt$ or $\dfrac{2}{x^3}+\dfrac{5}{x^6}dx=dt$

$\Rightarrow \dfrac{\dfrac{2}{x^3}+\dfrac{5}{x^6}}{\left(1+\dfrac{1}{x^2}+\dfrac{1}{x^5}\right)^3}dx=-\displaystyle\int \dfrac{1}{t^3}dt$

$=-\dfrac{t^{-2}}{-2}+C$

$=\dfrac{1}{2t^2}+C$

$=\dfrac{1}{2\left(1+\dfrac{1}{x^2}+\dfrac{1}{x^5}\right)^2}+C$

$=\dfrac{x^{10}}{2(x^5+x^3+1)^2}+C$.

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Single Correct Medium Published on 17th 09, 2020
Questions 203525
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