MCQ
$\int_0^a {{x^4}\sqrt {{a^2} - {x^2}} } \,dx = $
- A$\frac{\pi }{{32}}$
- ✓$\frac{\pi }{{32}}{a^6}$
- C$\frac{\pi }{{16}}{a^6}$
- D$\frac{\pi }{8}{a^6}$
$\Rightarrow dx = a\cos \theta \,\,d\theta $
Now $\int_0^a {{x^4}\sqrt {{a^2} - {x^2}} } dx = {a^6}\int_0^{\pi /2} {{{\sin }^4}\theta \cos \theta \cos \theta \,d\theta } $
$ = {a^6}\int_0^{\pi /2} {{{\sin }^4}\theta\, {{\cos }^2}\theta \,d\theta } $
$ = {a^6}\frac{{\Gamma \left( {\frac{5}{2}} \right).\Gamma \left( {\frac{3}{2}} \right)}}{{2\Gamma 4}} = \frac{\pi }{{32}}{a^6}$,
(Using gamma function).
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