If $\int\limits_0^1 \frac{a \sqrt{x} d x}{\left(x^{\frac{3}{2}}+1\right)^2}=1$ then a is equal to

3

The correct answer is Option (2) - 3

$\int\limits_0^1 \frac{a \sqrt{x} d x}{\left(x^{\frac{3}{2}}+1\right)^2}=1$

$y=\left(x^{\frac{3}{2}}+1\right)$

$dy=\frac{3}{2}\sqrt{x}dx$

$⇒\int\limits_1^2\frac{2a}{3}\frac{dy}{y^2}=1$

$x→0, y→1$

$x→1, y→2$

$⇒\frac{2a}{3}\left[\frac{-1}{y}\right]_1^2=1⇒\frac{a}{3}=1$

$⇒a=3$