Question

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

• A. $\frac{2}{3}si{n}^{-1}\left(\frac{x^{3/2}}{a^{3/2}}\right)+C$
• B. $\frac{1}{3}si{n}^{-1}\left(\frac{x^{3/2}}{a^{3/2}}\right)+C$
• C. $\frac{2}{3}si{n}^{-1}\left(\frac{x^{3/2}}{a^{3/2}}\right)$
• D. $\frac{2}{3}si{n}^{-1}\left(\frac{x^{3/2}}{a}\right)+C$

Answer 1

The correct option is A $\frac{2}{3}si{n}^{-1}\left(\frac{x^{3/2}}{a^{3/2}}\right)+C$

$I=\int \sqrt{\frac{x}{a^3-x^3}}dx=\int \frac{\sqrt{x}}{\sqrt{{\left(a^{3/2}\right)}^2-{\left(x^{3/2}\right)}^2}}dx$
Let $x^{3/2}=t$ or $\frac{3}{2}{x}^{1/2}dx=dt$ or $dx=\frac{2}{3\sqrt{x}}dt$
$\therefore I=\int \frac{2/3dt}{\sqrt{{\left(a^{3/2}\right)}^2-t^2}}dt=\frac{2}{3}\int \frac{dt}{\sqrt{{\left(a^{3/2}\right)}^2-t^2}}$
$=\frac{2}{3}si{n}^{-1}\left(\frac{t}{a^{3/2}}\right)+C=\frac{2}{3}si{n}^{-1}\left(\frac{x^{3/2}}{a^{3/2}}\right)+C$