Mathematics - Indefinite Integration Question with Solution | TestHub

The given integral can be written as

$I=\int \frac{3. x^{-3}+2x^{-5}}{{\left(2+3x^{-2}+x^{-4}\right)}^4}dx$

 (Dividing  $x^{16}$ to both numerator and denominator),

Let, $2+3x^{-2}+x^{-4}=t$

$\Rightarrow \left(-6x^{-3}-4x^{-5}\right)dx=dt$

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

$\therefore I=-\frac{1}{2}\int \frac{dt}{t^4}=-\frac{1}{2}\left(\frac{t^{-4+1}}{-4+1}\right)=\frac{1}{6}\left(\frac{1}{t^3}\right)+C$, where $C$ is the constant of integration.

$=\frac{1}{6{\left(2+3x^{-2}+x^{-4}\right)}^3}+C=\frac{x^{12}}{6{\left(2x^4+3x^2+1\right)}^3}+C$