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Integrate: $ \int \large\frac{dx}{x \sqrt {1-x^3}}$$= a \log \bigg| \large\frac{\sqrt {1-x^3}-1}{\sqrt {1-x^3}+1}\bigg|$ find the value of a

$(a)\;2 \qquad(b)\;3 \qquad(c)\;1 \qquad (d)\;\frac{2}{3}$

1 Answer

Suppose : $x^3 = \sin ^2 \theta$
By differentiate $3x^2 dx= 2 \sin \theta \cos \theta d \theta$
take L.H.S
=> $\int \large\frac{2 \sin \theta \cos \theta d \theta}{3 sin ^2 \theta \sqrt {1- \sin ^2 \theta}}$
=> $\int \large\frac{2}{3}$$ cosec \theta d \theta$
=> $ \large\frac{2}{3}$$ \int cosec \theta d \theta$
=> $ \large\frac{2}{3}$$ \log | cosec \theta - \cot \theta |+c$
=> $ \large\frac{2}{3}$$ \log \bigg| \large\frac{\cos \theta - \sin \theta}{\sin \theta \cos \theta} \bigg |$$+c$
To solve it we get,
$a= \large\frac{2}{3}$
Hence d is the correct answer.
answered Dec 23, 2013 by meena.p