Value of ∫ x^3√(1-x^2) dx is:

The correct answer is Option (3) → \( \displaystyle\frac{1}{5}(1-x^{2})^{\frac{5}{2}}-\displaystyle\frac{1}{3}(1-x^{2})^{\frac{3}{2}}+\mathrm{C} \): where C is an arbitrary constant

Let \( 1-x^{2}=t \). Then \( x\,dx=-\frac{dt}{2} \) and \( x^{2}=1-t \), and with \( x^{3}=x^{2}\cdot x \) the integral becomes

    \( \displaystyle \int x^{2}\sqrt{1-x^{2}}\;x\,dx=-\frac{1}{2}\int(1-t)\,t^{1/2}\,dt \)

Multiplying out and integrating:

    \( \displaystyle =-\frac{1}{2}\left[\frac{2}{3}t^{3/2}-\frac{2}{5}t^{5/2}\right] \)

    \( \displaystyle =\frac{1}{5}t^{5/2}-\frac{1}{3}t^{3/2}+\mathrm{C} \)

Replacing \( t=1-x^{2} \) gives Option (3).