Question

$\int \sqrt{\frac{x}{a^3-x^3}}dx=$

• A. $\frac{2}{3}{\cos }^{-1}(\frac{x}{a}{)}^{3/2}+C$
• B. $\frac{1}{3}{\sin }^{-1}(\frac{x}{a}{)}^{3/2}+c$
• C. $\frac{2}{3}{\sin }^{-1}(\frac{x}{a}{)}^{3/2}+c$
• D. $\frac{1}{3}{\cos }^{-1}(\frac{x}{a}{)}^{3/2}+c$

Answer 1

Answer: (C)

$\frac{2}{3}{\sin }^{-1}(\frac{x}{a}{)}^{3/2}+c$

$\int \sqrt{\frac{x}{a^3-x^3}}dx$
$\frac{1}{a}\int \frac{\sqrt{\frac{x}{a}}}{\sqrt{1-\frac{x^3}{a^3}}}dx$
$\frac{1}{a}\int \frac{2}{3}\frac{d(\frac{x}{a}{)}^{\frac{3}{2}}}{\sqrt{1-\left[\right(\frac{x}{a}{)}^{\frac{3}{2}}}{]}^2}$
$=\frac{2}{3}\times \frac{1}{a}\times a{\sin }^{-1}\left[\right(\frac{x}{a}{)}^{\frac{3}{2}}]+c$
$=\frac{2}{3}{\sin }^{-1}\left[\right(\frac{x}{a}{)}^{\frac{3}{2}}]+c$