Mathematics - Indefinite Integration Question with Solution | TestHub

Given,$I=\int \frac{2x^{12}+5x^9}{{\left(x^5+x^3+1\right)}^3}dx$

$=\int \frac{\left(2x^{12}+5x^9\right)dx}{x^{15}{\left(1+\frac{1}{x^2}+\frac{1}{x^5}\right)}^3}$

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

Let,$1+\frac{1}{x^2}+\frac{1}{x^5}=t$$\Rightarrow \left(-\frac{2}{x^3}-\frac{5}{x^6}\right)dx=dt$

$\Rightarrow \left(\frac{2}{x^3}+\frac{5}{x^6}\right)dx=-dt$

$I=-\int \frac{dt}{t^3}=\frac{1}{2 . t^2}$

$I=\frac{1}{2}\frac{1}{{\left(1+\frac{1}{x^2}+\frac{1}{x^5}\right)}^2}+c=\frac{1}{2}\frac{x^{10}}{{\left(x^5+x^3+1\right)}^2}+c$
Where$c$is the constant of integration.
$=\frac{x^{10}}{2{\left(x^5+x^3+1\right)}^2}+c$