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Question: The value of \(\int_{0}^{\pi/2}{\left( \sqrt{\sin\theta}\cos\theta \right)^{3}d\theta}\) is...

The value of ∫0π/2(sin⁡θcos⁡θ)3dθ\int_{0}^{\pi/2}{\left( \sqrt{\sin\theta}\cos\theta \right)^{3}d\theta} is

A

2/9

B

2/15

C

8/45

D

5/2

Answer

8/45

Explanation

Solution

∫0π/2(sin⁡θcos⁡θ)3dθ=∫0π/2sin⁡3/2θcos⁡3θdθ\int_{\mathbf{0}}^{\mathbf{\pi/2}}{\mathbf{(}\sqrt{\mathbf{\sin}\mathbf{\theta}}\mathbf{\cos}\mathbf{\theta}\mathbf{)}^{\mathbf{3}}\mathbf{d\theta =}\int_{\mathbf{0}}^{\mathbf{\pi/2}}{\mathbf{\sin}^{\mathbf{3/2}}\mathbf{\theta}\mathbf{\cos}^{\mathbf{3}}\mathbf{\theta}\mathbf{d\theta}}}

Applying gamma function,

∫0π/2sin⁡3/2θcos⁡3θdθ=Γ(32+12)Γ(3+12)2Γ(32+3+22)\int_{\mathbf{0}}^{\mathbf{\pi/2}}{\mathbf{\sin}^{\mathbf{3/2}}\mathbf{\theta}\mathbf{\cos}^{\mathbf{3}}\mathbf{\theta}\mathbf{d\theta}}\mathbf{=}\frac{\mathbf{\Gamma}\left( \frac{\frac{\mathbf{3}}{\mathbf{2}}\mathbf{+ 1}}{\mathbf{2}} \right)\mathbf{\Gamma}\left( \frac{\mathbf{3 + 1}}{\mathbf{2}} \right)}{\mathbf{2}\mathbf{\Gamma}\left( \frac{\frac{\mathbf{3}}{\mathbf{2}}\mathbf{+ 3 + 2}}{\mathbf{2}} \right)}

=Γ(5/4)Γ22Γ(13/4)=Γ(54)2.94.54.Γ(54)=845\mathbf{=}\frac{\mathbf{\Gamma(5/4)\Gamma 2}}{\mathbf{2\Gamma(13/4)}}\mathbf{=}\frac{\mathbf{\Gamma}\left( \frac{\mathbf{5}}{\mathbf{4}} \right)}{\mathbf{2.}\frac{\mathbf{9}}{\mathbf{4}}\mathbf{.}\frac{\mathbf{5}}{\mathbf{4}}\mathbf{.\Gamma}\left( \frac{\mathbf{5}}{\mathbf{4}} \right)}\mathbf{=}\frac{\mathbf{8}}{\mathbf{45}}.