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Thursday, June 24, 2021

Integral of the day

Integral involving the compostion of sin function with itself

We prove the following integral: π20sinsinxdx=113!!2+15!!217!!2+...
Proof
Recall the series expansion of sin(x) sinx=n=0(1)n(2n+1)!x2n+1xR
Therefore π20sinsinxdx=π20n=0(1)n(2n+1)!sin2n+1(x)dx=n=0(1)n(2n+1)!π20sin2n+1(x)dx=n=0(1)n(2n+1)!10w2n+1(1w2)12dw(wsinx)=n=0(1)n(2n+1)!10tn(1t)122dt(tw2)=n=0(1)n(2n+1)!B(n+1,12)2=n=0(1)n(2n+1)!Γ(n+1)Γ(12)2Γ(n+1+12)=n=0(1)n(2n+1)!n!π2Γ(n+1+12)(Γ(12)=π))
Now recall the Legrende duplication formula πΓ(2z)=22z1Γ(z)Γ(z+12)
Applying the formula to Γ(n+1+12) n=0(1)n(2n+1)!n!π2Γ(n+1+12)=n=0(1)n(2n+1)!22nn!Γ(n+1)Γ(2n+2)=n=0(1)n(2n+1)!22nn!2(2n+1)!=n=0(1)n22nn!2(2n+1)!2=n=0(1)n1(2n+1)!!2
π20sinsinxdx=n=0(1)n1(2n+1)!!2

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