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Please-Help-0-x-e-x-sinx-dx-

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Question Number 193278 by Nimnim111118 last updated on 09/Jun/23
Please Help...!! ∫^( ∞) _( 0) x.e^(−x) .sinx.dx
PleaseHelp…!!∫0∞x.e−x.sinx.dx
Answered by qaz last updated on 09/Jun/23
∫_0 ^∞ xe^(−x) sin xdx=−im∫_0 ^∞ xe^(−(1+i)x) dx =−im(1/((1+i)^2 ))=(1/2) −−−−− ∫_0 ^∞ xe^(−x) sin xdx=−DL{e^(−x) sin x}(s=0)=−D(1/((1+s)^2 +1))(s=0) =((2(1+s))/(((1+s)^2 +1)^2 ))(s=0)=(1/2)
∫0∞xe−xsinxdx=−im∫0∞xe−(1+i)xdx=−im1(1+i)2=12−−−−−∫0∞xe−xsinxdx=−DL{e−xsinx}(s=0)=−D1(1+s)2+1(s=0)=2(1+s)((1+s)2+1)2(s=0)=12
Commented by Nimnim111118 last updated on 10/Jun/23
Thank you sir. But, I dont clearly understand it.
Thankyousir.But,Idontclearlyunderstandit.

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