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Questions  >>  JEEMAIN and NEET  >>  Mathematics  >>  Class12  >>  Integral Calculus
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Q)

Integrate: $ \int \sqrt {\large\frac{x}{a^3-x^3}}$$dx$

$(a)\;\frac{2}{3} \sin ^{-1} \bigg( \frac{t}{a^{3/2}}\bigg ) +c \qquad(b)\;\frac{2}{3} \cos ^{-1} \bigg( \frac{t}{a^{3/2}}\bigg ) +c \qquad(c)\;\frac{2}{3} \tan ^{-1} \bigg( \frac{t}{a^{3/2}}\bigg ) +c \qquad (d)\;None$

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A)
$\int \large\frac{\sqrt x }{\sqrt {a^3-x^3}}$$dx$
$x^{3/2} x=t$
differentiate with x
$\large\frac{3}{2} $$x^{1/2} dx=dt$
=> $\int \large\frac{2}{3} \large\frac{dt}{\sqrt {(a^{3/2})^2 -(t)^2}}$
$\large\frac{2}{3} $$ \sin ^{-1} \bigg( \large\frac{t}{a^{3/2}}\bigg ) $$+c$
Hence a is the correct answer.
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