First slide

Methods of integration

Question

If $I = \int \frac{d x}{x^{3} {(1 + x^{6})}^{2 / 3}} = f (x) {(1 + x^{- 6})}^{1 / 3} + C,$ where $C$ is a constant of integration, then $f (x)$ is equal to:

Easy

A

$- \frac{1}{6}$
B

$- \frac{6}{x}$
C

$- \frac{x}{2}$
D

$- \frac{1}{2}$

Solution

$\begin{array}{l} I = \int \frac{d x}{x^{3} {(1 + x^{6})}^{2 / 3}} = \int \frac{d x}{x^{3} \cdot {(x^{6})}^{2 / 3} {(x^{- 6} + 1)}^{2 / 3}} \\ = \int \frac{x^{- 7}}{{(x^{- 6} + 1)}^{2 / 3}} d x \end{array}$
$= - \frac{1}{6} \cdot 3 {(x^{- 6} + 1)}^{1 / 3} + C$ [Put $x^{- 6} + 1 = t$ ]
$∴ f (x) = - 1 / 2$

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