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Question Number 156339 by lapache last updated on 10/Oct/21

∫_0 ^1 (x^(2n) /(arcsinx))dx=...???

∫01x2narcsinxdx=...???

Answered by ArielVyny last updated on 10/Oct/21

t=arcsinx→sint=x ∫_0 ^(π/2) ((sin^(2n) t)/t)costdt=∫_0 ^(π/2) ((sin^(2n) t)/t)Σ_(n≥0) (((−1)^n t^(2n) )/((2n)!)) Σ_(n≥0) (((−1)^n )/((2n)!))∫_0 ^(π/2) t^(2n) sin^(2n) tdt

t=arcsinx→sint=x∫0π2sin2nttcostdt=∫0π2sin2ntt∑n⩾0(−1)nt2n(2n)!∑n⩾0(−1)n(2n)!∫0π2t2nsin2ntdt

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