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Question Number 144174 by lapache last updated on 22/Jun/21
∫_0 ^π (sinx)^(2n) dx=....? ∀n∈N
$$\int_{\mathrm{0}} ^{\pi} \left({sinx}\right)^{\mathrm{2}{n}} {dx}=….?\:\:\:\forall{n}\in\mathbb{N} \\ $$
Commented by Willson last updated on 22/Jun/21
4^n sin^(2n) t = C_(2n) ^n +2Σ_(k=0) ^(n−1) (−1)^n C_(2n) ^(n−k) cos(2kt) 4^n ∫^( 𝛑) _( 0) sin^(2n) t dt= 𝛑C_(2n) ^n +2Σ_(k=0) ^(n−1) (−1)^k ∫^( 𝛑) _0 cos(2kt)dt_(0) ⇒ ∫^( π) _0 (sint)^(2n) dt= π((C_(2n) ^n /4^n ))
$$\mathrm{4}^{\mathrm{n}} \mathrm{sin}^{\mathrm{2n}} \mathrm{t}\:=\:\mathrm{C}_{\mathrm{2n}} ^{\mathrm{n}} +\mathrm{2}\underset{\mathrm{k}=\mathrm{0}} {\overset{\mathrm{n}−\mathrm{1}} {\sum}}\left(−\mathrm{1}\right)^{\mathrm{n}} \mathrm{C}_{\mathrm{2n}} ^{\mathrm{n}−\mathrm{k}} \mathrm{cos}\left(\mathrm{2kt}\right) \\ $$$$\mathrm{4}^{\mathrm{n}} \underset{\:\mathrm{0}} {\int}^{\:\boldsymbol{\pi}} \mathrm{sin}^{\mathrm{2n}} \mathrm{t}\:\mathrm{dt}=\:\boldsymbol{\pi}\mathrm{C}_{\mathrm{2n}} ^{\mathrm{n}} \:+\mathrm{2}\underset{\mathrm{0}} {\underbrace{\underset{\mathrm{k}=\mathrm{0}} {\overset{\mathrm{n}−\mathrm{1}} {\sum}}\left(−\mathrm{1}\right)^{\mathrm{k}} \underset{\mathrm{0}} {\int}^{\:\boldsymbol{\pi}} \mathrm{cos}\left(\mathrm{2kt}\right)\mathrm{dt}}}\: \\ $$$$\Rightarrow\:\underset{\mathrm{0}} {\int}^{\:\pi} \left(\mathrm{sint}\right)^{\mathrm{2n}} \mathrm{dt}=\:\pi\left(\frac{\mathrm{C}_{\mathrm{2n}} ^{\mathrm{n}} }{\mathrm{4}^{\mathrm{n}} }\right) \\ $$
Answered by Dwaipayan Shikari last updated on 22/Jun/21
∫_0 ^π (sinx)^(2n) dx=∫_(π/2) ^π (sinx)^(2n) dx+∫_0 ^(π/2) (sinx)^(2n) dx x=z−(π/2) ⇒∫_0 ^(π/2) (−sin((π/2)−z))^(2n) +∫_0 ^(π/2) sin^(2n) (x)dx =∫_0 ^(π/2) sin^(2n) (z)dz+∫_0 ^(π/2) sin^(2n) (x)dx=^(Dummy variable) 2∫_0 ^(π/2) sin^(2n) (x)dx Now ∫_0 ^(π/2) sin^(2a−1) (x)cos^(2b−1) (x)dx=((Γ(b)Γ(a))/(2Γ(b+a))) Here 2∫_0 ^(π/2) sin^(2n) (x)dx=((Γ((1/2))Γ(((2n+1)/2)))/(Γ(n+1)))=((√π)/(n!))Γ(n+(1/2))
$$\int_{\mathrm{0}} ^{\pi} \left({sinx}\right)^{\mathrm{2}{n}} {dx}=\int_{\frac{\pi}{\mathrm{2}}} ^{\pi} \left({sinx}\right)^{\mathrm{2}{n}} {dx}+\int_{\mathrm{0}} ^{\frac{\pi}{\mathrm{2}}} \left({sinx}\right)^{\mathrm{2}{n}} {dx} \\ $$$${x}={z}−\frac{\pi}{\mathrm{2}}\:\:\:\:\Rightarrow\int_{\mathrm{0}} ^{\frac{\pi}{\mathrm{2}}} \left(−{sin}\left(\frac{\pi}{\mathrm{2}}−{z}\right)\right)^{\mathrm{2}{n}} +\int_{\mathrm{0}} ^{\frac{\pi}{\mathrm{2}}} {sin}^{\mathrm{2}{n}} \left({x}\right){dx} \\ $$$$=\int_{\mathrm{0}} ^{\frac{\pi}{\mathrm{2}}} {sin}^{\mathrm{2}{n}} \left({z}\right){dz}+\int_{\mathrm{0}} ^{\frac{\pi}{\mathrm{2}}} {sin}^{\mathrm{2}{n}} \left({x}\right){dx}\overset{{Dummy}\:{variable}} {=}\mathrm{2}\int_{\mathrm{0}} ^{\frac{\pi}{\mathrm{2}}} {sin}^{\mathrm{2}{n}} \left({x}\right){dx} \\ $$$${Now}\:\int_{\mathrm{0}} ^{\frac{\pi}{\mathrm{2}}} {sin}^{\mathrm{2}{a}−\mathrm{1}} \left({x}\right){cos}^{\mathrm{2}{b}−\mathrm{1}} \left({x}\right){dx}=\frac{\Gamma\left({b}\right)\Gamma\left({a}\right)}{\mathrm{2}\Gamma\left({b}+{a}\right)} \\ $$$${Here}\:\mathrm{2}\int_{\mathrm{0}} ^{\frac{\pi}{\mathrm{2}}} {sin}^{\mathrm{2}{n}} \left({x}\right){dx}=\frac{\Gamma\left(\frac{\mathrm{1}}{\mathrm{2}}\right)\Gamma\left(\frac{\mathrm{2}{n}+\mathrm{1}}{\mathrm{2}}\right)}{\Gamma\left({n}+\mathrm{1}\right)}=\frac{\sqrt{\pi}}{{n}!}\Gamma\left({n}+\frac{\mathrm{1}}{\mathrm{2}}\right) \\ $$

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