Definite integral of the cube root of $x \ln (x)$ I was trying to solve the 2016 CSE question paper for Math Optional
$$I = \int_{0}^1 \left (x\log \left (\frac{1}{x}\right)\right)^{\frac{1}{3}} dx$$
In my attempt to find $I$, I tried to substitute 
$$t = \log \left (\frac{1}{x}\right) \\\ dt = \frac{x}{-x^2}$$
Even tried taking 
$$t = x\log \left (\frac{1}{x}\right)$$
and then tried by parts. But I am unable to make headway. Any idea what might work? How can I tackle these types of problems?

Take $x = e^{-3t/4}$
Then $\log\frac{1}{x} = 3t/4$
And $t$ varies from $0$ to $\infty$
Substitute these values in question
You'll get the answer in gamma function
