Mathematics - Definite Integration Question with Solution | TestHub

Using property

$I={\int }_{-1}^1(1-x{)}^{1/2}(1+x{)}^{3/2}dx$

$\Rightarrow 2I={\int }_{-1}^1(1+x{)}^{1/2}(1-x{)}^{1/2}[(1-x)+(1+x)]dx$

$\Rightarrow 2I=2{\int }_{-1}^1\sqrt{1-x^2}dx$

$\Rightarrow I=2{\int }_0^1\sqrt{1-x^2}dx(x=\sin \theta )\Rightarrow dx=\cos \theta d\theta$

$\Rightarrow I=2{\int }_0^{\pi /2}{\cos }^2\theta d\theta =\frac{\pi }{2}$