$\int \frac{(x - x^{5})^{1 / 5}}{x^{6}} d x$ is equal to

\frac{5}{24} (\frac{1}{x^{4}} - 1)^{6 / 5} + c

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\frac{5}{24} (1 - \frac{1}{x^{4}})^{6 / 5} + c

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\frac{- 5}{24} (\frac{1}{x^{4}} - 1)^{6 / 5} + c

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\frac{5}{24} (1 + \frac{1}{x^{4}})^{6 / 5} + c

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Solution

The correct option is C $\frac{- 5}{24} (\frac{1}{x^{4}} - 1)^{6 / 5} + c$
$I = \int \frac{{(x - x^{5})}^{1 / 5}}{x^{6}} d x$

$= \int \frac{{[x^{5} (\frac{1}{x^{4}} - 1)]}^{1 / 5}}{x^{6}} d x$

$= \int \frac{{(\frac{1}{x^{4}} - 1)}^{1 / 5}}{x^{5}} d x$

Put $\frac{1}{x^{4}} - 1 = t$

$\Rightarrow - \frac{4}{x^{5}} d x = d t$

So, $I = - \frac{1}{4} \int t^{1 / 5} d t$

$= - \frac{1}{4} \times \frac{5}{6} t^{6 / 5} + c$

$I = \frac{- 5}{24} {(\frac{1}{x^{4}} - 1)}^{6 / 5} + c$

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