The value of $\int \frac{2 x^{12}+5 x^9}{\left(x^5+x^3+1\right)^3} \mathrm{~d} x$ is equal to (where $C$ is arbitrary constant.)

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Aicharya · Accepted Answer

$\begin{aligned} & \int \frac{2 x^{12}+5 x^9}{\left(x^5+x^3+1ight)^3} \mathrm{~d} x=\int \frac{2 x^{12}+5 x^9}{x^{15}\left(1+\frac{1}{x^2}+\frac{1}{x^5}ight)^3} \mathrm{~d} x \ & =\int \frac{\frac{2}{x^3}+\frac{5}{x^6}}{\left(1+\frac{1}{x^2}+\frac{1}{x^5}ight)^3} \mathrm{~d} x \ & =\int \frac{-\mathrm{d} t}{t^3}=\frac{1}{2 t^2}+C\left[\operatorname{let} 1+\frac{1}{x^2}+\frac{1}{x^5}=tight] \ & =\frac{1}{2\left(1+\frac{1}{x^2}+\frac{1}{x^5}ight)^2}+C \ & =\frac{x^{10}}{2\left(x^5+x^3+1ight)^2}+C\end{aligned}$

The value of $\int \frac{2 x^{12}+5 x^9}{\left(x^5+x^3+1\right)^3} \mathrm{~d} x$ is equal to (where $C$ is…

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