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Question Number 117280 by mnjuly1970 last updated on 10/Oct/20

                           ... advanced  calculus...           prove  that::                   ∫_0 ^( 1) ln(Γ(x)).cos^2 (πx)dx                           =((ln(2π))/4)+(1/8)         m.n.1970

...advancedcalculus...provethat::01ln(Γ(x)).cos2(πx)dx=ln(2π)4+18m.n.1970

Answered by AbduraufKodiriy last updated on 10/Oct/20

∫_a ^( b) f(x)dx=∫_a ^( b) f(a+b−x)dx  I=∫_0 ^( 1) ln(𝚪(x))cos^2 (πx)dx=∫_0 ^( 1) ln(𝚪(1−x))cos^2 (π(1−x))dx=  =∫_0 ^( 1) ln(𝚪(1−x))cos^2 (πx)dx  Therefore: 2I=∫_0 ^( 1) (ln(𝚪(x))+ln(𝚪(1−x)))cos^2 (πx)dx ⇒  ⇒ I=(1/2)∫_0 ^( 1) ln(𝚪(x)𝚪(1−x))cos^2 (πx)dx=(1/2)∫_0 ^( 1) ln((π/(sin(πx))))cos^2 (πx)dx=  =((ln(π))/2)∫_0 ^( 1) cos^2 (πx)dx−(1/2)∫cos^2 (πx)ln(sin(πx))dx;  I_1 =∫_0 ^( 1) cos^2 (πx)ln(sin(πx))dx ⇒ I=((ln(π))/4)−(1/2)I_1   I_1 = determinant (((πx=t ⇒ x=(t/π))),((dx=(1/π)dt)))=(1/π)∫_0 ^( π) cos^2 (t)ln(sin(t))dt=  =(1/(2π))∫_0 ^( π) ln(sin(t))dt+(1/(2π))∫_0 ^( π) cos(2t)ln(sin(t))dt; We know: ∫_0 ^( π) ln(sin(t))dt=−πln2  I_1 =−(1/2)ln(2)+(1/2)∫_0 ^( π) cos(2t)ln(sin(t))dt=−(1/2)ln2+(((sin(2t)ln(sin(t)))/(4π))−(t/(4π))−((sin(2t))/(8π)))∣_0 ^π =  =−(1/2)ln(2)−(1/4) ⇒ I=((ln(π))/4)+((ln(2))/4)+(1/8)=((ln(2π))/4)+(1/8)

abf(x)dx=abf(a+bx)dxI=01ln(Γ(x))cos2(πx)dx=01ln(Γ(1x))cos2(π(1x))dx==01ln(Γ(1x))cos2(πx)dxTherefore:2I=01(ln(Γ(x))+ln(Γ(1x)))cos2(πx)dxI=1201ln(Γ(x)Γ(1x))cos2(πx)dx=1201ln(πsin(πx))cos2(πx)dx==ln(π)201cos2(πx)dx12cos2(πx)ln(sin(πx))dx;I1=01cos2(πx)ln(sin(πx))dxI=ln(π)412I1I1=|πx=tx=tπdx=1πdt|=1π0πcos2(t)ln(sin(t))dt==12π0πln(sin(t))dt+12π0πcos(2t)ln(sin(t))dt;Weknow:0πln(sin(t))dt=πln2I1=12ln(2)+120πcos(2t)ln(sin(t))dt=12ln2+(sin(2t)ln(sin(t))4πt4πsin(2t)8π)0π==12ln(2)14I=ln(π)4+ln(2)4+18=ln(2π)4+18

Commented by AbduraufKodiriy last updated on 10/Oct/20

Commented by mnjuly1970 last updated on 10/Oct/20

than you mr abduraufk..

thanyoumrabduraufk..

Commented by mnjuly1970 last updated on 10/Oct/20

javob=javab=answer...

javob=javab=answer...

Commented by mnjuly1970 last updated on 10/Oct/20

mercey aghaye abduraufk

merceyaghayeabduraufk

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