MCQ
$\int_{\,0}^{\,\pi /2} {{{\left( {\sqrt {\sin \theta } \cos \theta } \right)}^3}d\theta } = . . . ..$
- A$2/9$
- B$2/15$
- ✓$8/45$
- D$5/2$
Applying gamma function,
$\int_0^{\pi /2} {{{\sin }^{3/2}}\theta {{\cos }^3}\theta d\theta } $
$ = \frac{{\Gamma \left( {\frac{{\frac{3}{2} + 1}}{2}} \right)\Gamma \left( {\frac{{3 + 1}}{2}} \right)}}{{2\Gamma \left( {\frac{{\frac{3}{2} + 3 + 2}}{2}} \right)}}$
$ = \frac{{\Gamma (5/4)\,\,\Gamma \,2}}{{2\Gamma \,(13/4)}}$
$ = \frac{{\Gamma \,\left( {\frac{5}{4}} \right)}}{{2.\frac{9}{4}.\frac{5}{4}.\Gamma \left( {\frac{5}{4}} \right)}}$$ = \frac{8}{{45}}$.
Generate a complete, print-ready paper with questions like this in minutes — across 16+ boards, with answer keys.