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Question Number 139353 by mohammad17 last updated on 26/Apr/21
∫_0 ^( (π/2)) sin^6 θ cos^4 θ dθ
$$\int_{\mathrm{0}} ^{\:\frac{\pi}{\mathrm{2}}} {sin}^{\mathrm{6}} \theta\:{cos}^{\mathrm{4}} \theta\:{d}\theta \\ $$
Answered by Dwaipayan Shikari last updated on 26/Apr/21
∫_0 ^(π/2) sin^(2α−1) θ cos^(2β−1) θ dθ=((Γ(α)Γ(β))/(2Γ(α+β)))  ∫_0 ^(π/2) sin^6 θ cos^4 θ dθ=((Γ((7/2))Γ((5/2)))/(2Γ(6)))=(((5/2).(3/2).(1/2).(3/2).(1/2)Γ^2 ((1/2)))/(240))  =(3/2^9 )π=((3π)/(512))
$$\int_{\mathrm{0}} ^{\frac{\pi}{\mathrm{2}}} {sin}^{\mathrm{2}\alpha−\mathrm{1}} \theta\:{cos}^{\mathrm{2}\beta−\mathrm{1}} \theta\:{d}\theta=\frac{\Gamma\left(\alpha\right)\Gamma\left(\beta\right)}{\mathrm{2}\Gamma\left(\alpha+\beta\right)} \\ $$$$\int_{\mathrm{0}} ^{\frac{\pi}{\mathrm{2}}} {sin}^{\mathrm{6}} \theta\:{cos}^{\mathrm{4}} \theta\:{d}\theta=\frac{\Gamma\left(\frac{\mathrm{7}}{\mathrm{2}}\right)\Gamma\left(\frac{\mathrm{5}}{\mathrm{2}}\right)}{\mathrm{2}\Gamma\left(\mathrm{6}\right)}=\frac{\frac{\mathrm{5}}{\mathrm{2}}.\frac{\mathrm{3}}{\mathrm{2}}.\frac{\mathrm{1}}{\mathrm{2}}.\frac{\mathrm{3}}{\mathrm{2}}.\frac{\mathrm{1}}{\mathrm{2}}\Gamma^{\mathrm{2}} \left(\frac{\mathrm{1}}{\mathrm{2}}\right)}{\mathrm{240}} \\ $$$$=\frac{\mathrm{3}}{\mathrm{2}^{\mathrm{9}} }\pi=\frac{\mathrm{3}\pi}{\mathrm{512}} \\ $$
Answered by ajfour last updated on 26/Apr/21
I=∫_0 ^( π/2) sin^6 θcos^4 θdθ    =∫_0 ^( π/2) cos^6 θsin^4 θdθ  2I=∫_0 ^( π/2) sin^4 θcos^4 θdθ  32I=∫_0 ^( π/2) sin^4 2θdθ  128I=∫_0 ^( π/2) (1−cos 4θ)^2 dθ  256I=∫_0 ^( π/2) (2−4cos 4θ+1+cos 8θ)dθ  256I=((3π)/2)  I=((3π)/(512))
$${I}=\int_{\mathrm{0}} ^{\:\pi/\mathrm{2}} \mathrm{sin}\:^{\mathrm{6}} \theta\mathrm{cos}\:^{\mathrm{4}} \theta{d}\theta \\ $$$$\:\:=\int_{\mathrm{0}} ^{\:\pi/\mathrm{2}} \mathrm{cos}\:^{\mathrm{6}} \theta\mathrm{sin}\:^{\mathrm{4}} \theta{d}\theta \\ $$$$\mathrm{2}{I}=\int_{\mathrm{0}} ^{\:\pi/\mathrm{2}} \mathrm{sin}\:^{\mathrm{4}} \theta\mathrm{cos}\:^{\mathrm{4}} \theta{d}\theta \\ $$$$\mathrm{32}{I}=\int_{\mathrm{0}} ^{\:\pi/\mathrm{2}} \mathrm{sin}\:^{\mathrm{4}} \mathrm{2}\theta{d}\theta \\ $$$$\mathrm{128}{I}=\int_{\mathrm{0}} ^{\:\pi/\mathrm{2}} \left(\mathrm{1}−\mathrm{cos}\:\mathrm{4}\theta\right)^{\mathrm{2}} {d}\theta \\ $$$$\mathrm{256}{I}=\int_{\mathrm{0}} ^{\:\pi/\mathrm{2}} \left(\mathrm{2}−\mathrm{4cos}\:\mathrm{4}\theta+\mathrm{1}+\mathrm{cos}\:\mathrm{8}\theta\right){d}\theta \\ $$$$\mathrm{256}{I}=\frac{\mathrm{3}\pi}{\mathrm{2}} \\ $$$${I}=\frac{\mathrm{3}\pi}{\mathrm{512}} \\ $$

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