First slide

Methods of integration

Question

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

Moderate

A

$\frac{4}{15} {(1 - \frac{1}{x^{3}})}^{5 / 4} + C$
B

$\frac{4}{5} {(1 - \frac{1}{x^{3}})}^{5 / 4} + C$
C

$\frac{4}{15} {(1 + \frac{1}{x^{3}})}^{5 / 4} + C$
D

$None of these$

Solution

$\int \frac{{(x^{4} - x)}^{1 / 4}}{x^{5}} d x = \int \frac{1}{x^{4}} {(1 - \frac{1}{x^{3}})}^{1 / 4} d x$
$Putting 1 - \frac{1}{x^{3}} = t, we get$
$I = \frac{1}{3} \int t^{1 / 4} d t = \frac{4}{15} t^{5 / 4} + C = \frac{4}{15} {(1 - \frac{1}{x^{3}})}^{5 / 4} + C$

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