Integral involving the compostion of sin function with itself
We prove the following integral: ∫π20sinsinxdx=1−13!!2+15!!2−17!!2+...
Proof
Recall the series expansion of sin(x) sinx=∞∑n=0(−1)n(2n+1)!x2n+1∀x∈R
Therefore
∫π20sinsinxdx=∫π20∞∑n=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(1−w2)−12dw(w↦sinx)=∞∑n=0(−1)n(2n+1)!∫10tn(1−t)−122dt(t↦w2)=∞∑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)=22z−1Γ(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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