I=∫dxx21+x545 = ∫dxx61x5+145 Let t=1+1x5⇒dt=−5dxx6∴I=−15∫dtt4/5 =−t1/5+C =−1+1x515+C =−x5+115x+C

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

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$- \frac{{(1 + x^{5})}^{\frac{1}{5}}}{5 x} + c$

$\frac{{(1 + x^{5})}^{\frac{1}{5}}}{5 x} + c$

$- \frac{{(1 + x^{5})}^{\frac{1}{5}}}{x} + c$

$\frac{{(1 + x^{5})}^{\frac{1}{5}}}{x} + c$

answer is C.

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Detailed Solution

$\begin{array}{l} I = \int \frac{d x}{x^{2} {(1 + x^{5})}^{\frac{4}{5}}} = \int \frac{d x}{x^{6} {(\frac{1}{x^{5}} + 1)}^{\frac{4}{5}}} \\ Let t = 1 + \frac{1}{x^{5}} \Rightarrow d t = - \frac{5 d x}{x^{6}} \\ ∴ I = - \frac{1}{5} \int \frac{d t}{t^{4 / 5}} \\ = - t^{1 / 5} + C \\ = - {(1 + \frac{1}{x^{5}})}^{\frac{1}{5}} + C \\ = - \frac{{(x^{5} + 1)}^{\frac{1}{5}}}{x} + C \end{array}$