Writing complicated integral in terms of the Gamma function

1. Apr 12, 2015

davidbenari

1. The problem statement, all variables and given/known data
Write $\int_{0}^{1}x^2(ln\frac{1}{x})^3 dx$ in terms of the gamma function

2. Relevant equation
$\Gamma(p+1)=p\Gamma(p)$
3. The attempt at a solution
Say $x=e^{-u}$ one would eventually obtain the integral

$\int_{-\infty}^{0} u^3 e^{-u} du$

STEPS:
$x=e^{-u}$ $e^{u}=1/x$ $u=ln(1/x)$ $du=xdx$

$\int_{0}^{1}x^2(ln\frac{1}{x})^3 dx=\int_{0}^{-\infty} e^{-u} x u^3 \frac{du}{x}=\int_{-\infty}^{0} u^3 e^{-u} du$

Which wants to look like a gamma function but isnt because of the limits.

Last edited: Apr 12, 2015
2. Apr 12, 2015

Staff: Mentor

I don't know what went wrong with the substitution (can you show the steps?), but the last integral diverges.

3. Apr 12, 2015

davidbenari

4. Apr 12, 2015

davidbenari

5. Apr 13, 2015

Staff: Mentor

I don't think that is right.
And your integral limits look wrong, too.
x=1 corresponds to u=0, but x=0 corresponds to a different value for u.

6. Apr 13, 2015

davidbenari

Yeah I've corrected it now. Thanks!