First slide

Methods of integration

Question

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

Moderate

A

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

$\frac{5}{4} {(1 - \frac{1}{x^{4}})}^{6 / 5} + C$
C

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

none of these

Solution

Let
$\begin{array}{l} I = \int \frac{{(x - x^{5})}^{1 / 5}}{x^{6}} d x = \int {(\frac{1}{x^{4}} - 1)}^{1 / 5} \frac{1}{x^{5}} d x \\ \Rightarrow I = - \frac{1}{4} \int {(\frac{1}{x^{4}} - 1)}^{1 / 5} (\frac{- 4}{x^{5}}) d x = - \frac{1}{4} \int {(\frac{1}{x^{4}} - 1)}^{1 / 5} d (\frac{1}{x^{4}} - 1) \\ = I = - \frac{5}{24} {(\frac{1}{x^{4}} - 1)}^{6 / 5} + C \end{array}$

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