First slide

Methods of integration

Question

$\int \sqrt{\frac{x^{4}}{a^{6} - x^{6}}} d x is equal to$

Moderate

A

$\frac{1}{3} \sin^{- 1} (\frac{x^{3}}{a^{3}}) + C$
B

$\frac{1}{3} \sin^{- 1} (\frac{x^{3}}{a}) + C$
C

$\frac{1}{3} \sin^{- 1} (\frac{x^{3}}{a^{6}}) + C$
D

$\frac{- 1}{3} \sin^{- 1} (\frac{x^{3}}{a^{3}}) + C$

Solution

$Let I = \int \sqrt{\frac{x^{4}}{a^{6} - x^{6}}} d x$
$I = \int \frac{x^{2}}{\sqrt{(a^{6}) - {(x^{3})}^{2}}} d x$
$Put x^{3} = t$
$\begin{aligned} x^{2} d x = \frac{1}{3} d t \\ I = \int \frac{\frac{1}{3} d t}{\sqrt{{(a^{3})}^{2} - t^{2}}} \\ = \frac{1}{3} \sin^{- 1} (\frac{t}{a^{3}}) + C \\ = \frac{1}{3} \sin^{- 1} (\frac{x^{3}}{a^{3}}) + C \end{aligned}$

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