Mathematics - Indefinite Integration Question with Solution | TestHub

We have, $\int \frac{dx}{x^3{\left(1+x^6\right)}^{\frac{2}{3}}}=xf\left(x\right){\left(1+x^6\right)}^{\frac{1}{3}}+C$

Now, $I=\int \frac{dx}{x^7{\left(\frac{1}{x^6}+1\right)}^{\frac{2}{3}}}$

Put, $t=\frac{1}{x^6}+1$$\Rightarrow dt=-\frac{6}{x^7}dx$

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

$\therefore f\left(x\right)=-\frac{1}{2x^3}$