Question #092c3

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Answer 1 · 2017-11-05T16:39:34

#int_0^1 x^3*[Ln(1/x)]^3*dx#=#3/128#

#int_0^1 x^3*[Ln(1/x)]^3*dx#

After using #u=Ln(1/x)=-Lnx#, #Lnx=-u#, #x=e^(-u)# and #dx=-e^(-u)*du# transforms, this integral became,

#int_oo^0 -u^3*e^(-4u)*du#

#=int_0^oo u^3*e^(-4u)*du#

After using tabular integration,

#=int_0^oo u^3*e^(-4u)*du#

=#[u^3*(-1/4e^(-4u))-3u^2*1/16e^(-4u)+6u*(-1/64*e^(-4u))-6*1/256e^(-4u)]_0^oo#

=#[-1/128e^(-4u)*(32u^3+24u^2+12u+3)]_0^oo#

=#3/128#