If $\int \frac{d x}{x^{3}\left(1+x^{6}\right)^{2 / 3}}=x f(x)\left(1+x^{6}\right)^{\frac{1}{3}}+C$, where $\mathrm{C}$ is a
constant of integration, then the function $f(x)$ is equal to :
Correct Option: , 4
Let, $\int \frac{d x}{x^{3}\left(1+x^{6}\right)^{\frac{2}{3}}}=\int \frac{d x}{x^{7}\left(1+x^{-6}\right)^{\frac{2}{3}}}$
Put $1+x^{-6}=t^{3} \quad \Rightarrow-6^{-7} d x=3 t^{2} d t \Rightarrow \frac{d x}{x^{7}}=\left(-\frac{1}{2}\right) t^{2} d t$
Now, $I=\int\left(-\frac{1}{2}\right) \frac{t^{2} d t}{t^{2}}=-\frac{1}{2} t+C$
$=-\frac{1}{2}\left(1+x^{-6}\right)^{\frac{1}{3}}+C=-\frac{1}{2} \frac{\left(1+x^{6}\right)^{\frac{1}{3}}}{x^{2}}+C$
$=-\frac{1}{2 x^{3}} x\left(1+x^{6}\right)^{\frac{1}{3}}+C$
Hence, $f(x)=-\frac{1}{2 x^{3}}$
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