If where C is a constant of integration, then the function f (x) is equal to

Engineering

Mathematics

Integration of Miscellaneous Functions

Question

If $\int \frac{dx}{x^{3} {(1 + x^{6})}^{2 / 3}} = xf (x) {(1 + x^{6})}^{1 / 3} + C$

where C is a constant of integration, then the function f (x) is equal to

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

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

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

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

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

$= \int \frac{dx}{x^{7} {(1 + \frac{1}{x^{6}})}^{2 / 3}}$

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

$I = \frac{1}{6} \int \frac{- dt}{t^{2 / 3}} = \frac{- 1}{2} {[1 + \frac{1}{x^{6}}]}^{1 / 3} + C$

$= \frac{- 1}{2} \frac{{(1 + x^{6})}^{1 / 3}}{x^{2}} = xf (x) {(1 + {x^{6}}^{})}^{1 / 3} + C$

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