The correct option is C 2/3√(a^2 x^3)+c x^3=t 3x^2.dx=dt x^2.dx=1/3dt ∫1/3dt√(a^2 t)=13dt√(a^2 t) =13(2)√(a^2 t)( 1)+c = 23 ...

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Special Integrals - 1

∫x2/√a2-x3dX=

Question

$\int \frac{x^{2}}{\sqrt{a^{2} - x^{3}}} d_{X} =$

2 \sqrt{a^{2} - x^{3}} + c

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\frac{1}{\sqrt{a^{2} - x^{2}}} + c

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- \frac{2}{3} \sqrt{a^{2} - x^{3}} + c

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\frac{1}{3} \sqrt{a^{2} - x^{2}} + c

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Solution

The correct option is C $- \frac{2}{3} \sqrt{a^{2} - x^{3}} + c$
$x^{3} = t$
$3 x^{2} . d x = d t$
$x^{2} . d x = 1 / 3 d t$
$\int \frac{1 / 3 d t}{\sqrt{a^{2} - t}} = \frac{1}{3} \frac{d t}{\sqrt{a^{2} - t}}$
$= \frac{1}{3} (2) \sqrt{a^{2} - t} (- 1) + c$
$= - \frac{2}{3} \sqrt{a^{2} - t} + c$
$= - \frac{2}{3} \sqrt{a^{2} - x^{3}} + c$
$\int \frac{x^{2}}{\sqrt{a^{2} - x^{3}}} d x = - \frac{2}{3} \sqrt{a^{2} - x^{3}} + c$

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