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Prove-that-n-IN-1-0-t-sin-2n-lnt-dt-1-1-e-2pi-pi-0-e-2t-sin-2n-t-dt-

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Question Number 194868 by Erico last updated on 17/Jul/23
Prove that ∀n∈IN ∫^( 1) _( 0) t sin^(2n) (lnt)dt= (1/(1−e^(−2π) )) ∫^( π) _( 0) e^(−2t) sin^(2n) (t)dt
Provethat∀n∈IN∫01tsin2n(lnt)dt=11−e−2π∫0πe−2tsin2n(t)dt
Answered by witcher3 last updated on 17/Jul/23
ln(t)=−x ⇒∫_0 ^∞ e^(−2x) sin^(2n) (x)dx =Σ_(k≥0) ∫_(kπ) ^((k+1)π) e^(−2x) sin^(2n) (x)dx x→kπ+t =Σ_(k≥0) ∫_0 ^π e^(−2kπ−2t) sin^(2n) (kπ+t)dt =Σ_(k≥0) e^(−2kπ) ∫_0 ^π e^(−2t) sin^(2n) (t)dt =∫_0 ^π e^(−2t) sin^(2n) (t)dt.Σ_(k≥0) e^(−2kπ) =(1/(1−e^(−2π) ))∫_0 ^π e^(−2t) sin^(2n) (t)dt
ln(t)=−x⇒∫0∞e−2xsin2n(x)dx=∑k⩾0∫kπ(k+1)πe−2xsin2n(x)dxx→kπ+t=∑k⩾0∫0πe−2kπ−2tsin2n(kπ+t)dt=∑k⩾0e−2kπ∫0πe−2tsin2n(t)dt=∫0πe−2tsin2n(t)dt.∑k⩾0e−2kπ=11−e−2π∫0πe−2tsin2n(t)dt

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