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

$\dfrac{2}{3}\sin^{-1}\!\left(\dfrac{x^{3/2}}{a^{3/2}}\right)+C$

CBSE Class 12 · Indefinite IntegralsLong Answer

Find

\int \frac{x}{a ^{3} - x ^{3}} d x

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

Rewrite

a^{3} - x^{3} = (a^{3/2})^{2} - (x^{3/2})^{2}

. Put

t = x^{3/2}

, so

d t = \frac{3}{2} x^{1/2} d x

, i.e.

x d x = \frac{2}{3} d t

. Then

\int \frac{x d x}{( a ^{3/2} ) ^{2} - ( x ^{3/2} ) ^{2}} = \frac{2}{3} \int \frac{d t}{( a ^{3/2} ) ^{2} - t ^{2}} = \frac{2}{3} sin^{- 1} (\frac{t}{a ^{3/2}}) + C .

Restoring

t = x^{3/2}

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

CBSE Class 12 Board · Subjective