Find: dx x^ 3 (x^ 5 + 1)^ 3/5 - Vidyadip

Question

Find:
$\int\frac{\text{dx}}{\text{x}^{3}(\text{x}^{5} + 1)^{3/5}}$

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Answer

$\text{I} = \int\frac{\text{dx}}{\text{x}^{3}(\text{x}^{5} + 1)^{3/5}}$
$= \int\frac{\text{dx}}{\text{x}^{3}.\text{x}^{3}\bigg(1 + \frac{1}{\text{x}^{5}}\bigg)^{3/5}}$
$\text{Put 1} + \frac{1}{\text{x}^{5}} = \text{t}$
$\Rightarrow \frac{\text{dx}}{\text{x}^{6}} = -\frac{\text{dt}}{5}$
$\therefore \text{I} = -\frac{1}{5} \int\text{t}^{-3/5} \text{dt} = -\frac{1}{2} \text{t}^{2/5} + \text{C}$
$= -\frac{1}{2}\bigg(1 + \frac{1}{\text{x}^{5}}\bigg)^{2/5}+\text{C}$

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