# If $\int \frac{dx}{x^{3}\left ( 1+x^{6} \right )^{\frac{2}{3}}}=xf(x)(1+x^{6})^{\frac{1}{3}}+C$Where C is a constant of integration, then the function f(x) is equal to: Option 1) $\frac{3}{x^{2}}$ Option 2) $-\frac{1}{6x^{3}}$ Option 3) $-\frac{1}{2x^{2}}$ Option 4) $-\frac{1}{2x^{3}}$

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

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

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

$\therefore -\int \frac{1}{6}\frac{1}{t^{\frac{2}{3}}}dt = - \frac{1}{6}\times \frac{t^{\frac{2}{3}}}{\frac{1}{3}}+C$

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

Compare with $xf(x)(1+x^{6})^{\frac{1}{3}}$

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

Correct option (4)

Option 1)

$\frac{3}{x^{2}}$

Option 2)

$-\frac{1}{6x^{3}}$

Option 3)

$-\frac{1}{2x^{2}}$

Option 4)

$-\frac{1}{2x^{3}}$

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