MCQ
$\int_0^a {{x^2}{{({a^2} - {x^2})}^{3/2}}dx = } $
- ✓$\frac{{\pi {a^6}}}{{32}}$
- B$\frac{{2{a^5}}}{{15}}$
- C$\frac{{{a^6}}}{{32}}$
- DNone of these
Put $x = a\sin \theta \Rightarrow dx = a\cos \theta \,d\theta $
$I = \int_0^{\pi /2} {{a^2}{{\sin }^2}\theta .{a^3}{{\cos }^3}\theta .a\cos \theta \,d\theta } $
$ = {a^6}\int_0^{\pi /2} {{{\sin }^2}\theta {{\cos }^4}\theta \,d\theta= {a^6}\frac{{\Gamma \frac{3}{2}.\,\Gamma \frac{5}{2}}}{{2.\Gamma \frac{8}{2}}}} $
$= {a^6}\frac{{\frac{1}{2}.\sqrt \pi .\frac{3}{2}.\frac{1}{2}.\sqrt \pi }}{{2.3.2.1}} $
$= \frac{{\pi {a^6}}}{{32}}$.
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