First slide

Methods of integration

Question

$\int \frac{{(x^{4} - x)}^{\frac{1}{4}}}{x^{5}} d x = ......$

Easy

A

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

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

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

$\frac{4}{5} {(1 + \frac{1}{x^{3}})}^{\frac{5}{4}} + c$

Solution

$\int \frac{x {(1 - \frac{1}{x^{3}})}^{\frac{1}{4}}}{x^{5}} d x p u t 1 - \frac{1}{x^{3}} = t$
   $= \int {(1 - \frac{1}{x^{3}})}^{\frac{1}{4}} \frac{d x}{x^{4}} \frac{d x}{x^{4}} = \frac{d t}{3}$
$= \frac{+ 1}{3} \int t^{\frac{1}{4}} d t = \frac{1}{3} \frac{t^{1 / 4 + 1}}{\frac{1}{4} + 1} C$            $= \frac{4}{15} {(1 - \frac{1}{x^{3}})}^{\frac{5}{4}} + c$

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