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$\int \frac{d x}{x^{1 / 5} {(1 + x^{4 / 5})}^{1 / 2}} =$

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$\sqrt{1 + x^{4 / 5}} + c$

$\frac{5}{2} \sqrt{1 + x^{4 / 5}} + c$

$x^{4 / 5} \sqrt{1 + x^{4 / 5}} + c$

$\frac{2}{3} \sqrt{1 + x^{4 / 5}} + c$

answer is B.

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Detailed Solution

$\int \frac{d x}{x^{1 / 5} {(1 + x^{4 / 5})}^{1 / 2}}$

$p u t 1 + x^{4 / 5} = t$

$\Rightarrow \frac{4}{5} x^{- 1 / 5} d x = d t$

$\Rightarrow \frac{d x}{x^{1 / 5}} = \frac{5}{4} d t$

$= \frac{5}{4} \int \frac{d t}{t^{1 / 2}}$

$= \frac{5}{4} \frac{t^{1 / 2}}{1 / 2} + C$

$= \frac{5}{2} \sqrt{1 + x^{4 / 5}} + C$

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