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Question

Question: \[\int_{0}^{a}{x^{4}\sqrt{a^{2} - x^{2}}}dx =\]...

∫0ax4a2−x2dx=\int_{0}^{a}{x^{4}\sqrt{a^{2} - x^{2}}}dx =

A

π32\frac{\pi}{32}

B

π32a6\frac{\pi}{32}a^{6}

C

π16a6\frac{\pi}{16}a^{6}

D

π8a6\frac{\pi}{8}a^{6}

Answer

π32a6\frac{\pi}{32}a^{6}

Explanation

Solution

Put x=asin⁡θ⇒dx=acos⁡θdθx = a\sin\theta \Rightarrow dx = a\cos\theta d\theta

Now ∫0ax4a2−x2dx=a6∫0π/2sin⁡4θcos⁡θcos⁡θdθ\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}

=a6∫0π/2sin⁡4θcos⁡2θdθ=a6Γ(52).Γ(32)2Γ4=π32a6= 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).