Question

Question #092c3

Answer 1

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

Explanation:

`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`