First slide

Methods of integration

Question

$\int \frac{1}{x^{4} \sqrt[4]{x^{4} + 1}} d x =$

Moderate

A

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

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

$\frac{2}{3} {(1 + \frac{1}{x^{4}})}^{3 / 4} + C$
D

$\frac{- 2}{3} {(1 + \frac{1}{x^{4}})}^{3 / 4} + C$

Solution

$\int \frac{x^{- 5}}{\sqrt[4]{1 + \frac{1}{x^{4}}}} d x$
$1 + \frac{1}{x^{4}} = t \Rightarrow - 4. x^{- 5} d x = d t$

$- \frac{1}{4} \int \frac{1}{{(t)}^{\frac{1}{4}}} d t = - \frac{1}{4} \frac{t^{\frac{3}{4}}}{(\frac{3}{4})} + C$

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

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