Mathematics

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


ANSWER

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


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
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