Question

If $\int \frac{\sqrt{x}}{\sqrt{a^3-x^3}}dx=m{\sin }^{-1}{\left(\frac{x}{a}\right)}^n+c$, then

• A. $m=n$
• B. $m=-n$
• C. $m=1/n$
• D. $m=-1/n$

Answer 1

Answer: (C)

$m=1/n$

$I=\int \frac{\sqrt{x}}{\sqrt{a^3-x^3}}dx=msi{n}^{-1}(\frac{x}{a}{)}^n+c$

Substitute $x=asi{n}^{2/3}t$

i.e, $t=si{n}^{-1}(\frac{x}{a}{)}^{3/2}$

$dx=\frac{2}{3}asi{n}^{-1/3}tcost$ dt

$I=\int \frac{2}{3}\frac{a^{1/2}.si{n}^{1/3}t.a.si{n}^{-1/3}t.cost.dt}{a^{3/2}cost}$

$I=\frac{2}{3}t+c$

$I=\frac{2}{3}si{n}^{-1}(\frac{x}{a}{)}^{3/2}+c$

$\therefore m=\frac{1}{n}$