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Q.
If $\int \frac{dx}{x^{3}\left(1+x^{6}\right)^{\frac{2}{3}}}=f \left(x\right)\left(1+x ^{6}\right)^{\frac{1}{3}}+C$, where C is a constant of integration, then the function $f \left(x\right)$ is equal to-
$\int \frac{dx}{x^{3}\left(1+x^{6}\right)^{\frac{2}{3}}}= x f \left(x\right)\left(1+x^{6}\right)^{\frac{1}{3}} +c$
$\int \frac{dx}{x^{7}\left(\frac{1}{x^{6}}+1\right)^{\frac{2}{3}}}= x f \left(x\right)\left(1+x^{6}\right)^{\frac{1}{3}} +c$
Let $t=\frac{1}{x^{6}}+1$
$dt=\frac{-6}{x^{7}}dx$
$=- \frac{1}{6}\int \frac{dt}{t^{\frac{2}{3}}}=- \frac{1}{2} t^{\frac{1}{3}}$
$=- \frac{1}{2} \left(\frac{1}{x^{6}}+1\right)^{\frac{1}{3}} =- \frac{1}{2} \frac{\left(1+x^{6}\right)^{\frac{1}{3}}}{x^{2}}$
$\therefore f \left(x\right)=- \frac{1}{2x^{3}}$