Evaluate $\int_0^1(x\ln(x))^{50}dx$ Evaluate $$\int_0^1(x\ln(x))^{50} dx.$$


Here are my steps so far using differentiation under the integral sign:

$$I(t) = \int_0^1(x\ln(x))^t dx$$
$$I'(t) = \frac d{dt}\int_0^1(x\ln(x))^t dx = \int_0^1\frac \partial{\partial t}(x\ln(x))^t dx = \int_0^1(x\ln(x))^t\ln(x\ln(x)) dx$$

I can't find a way to continue so hints are appreciated.
Hint. Following your approach by differentiation under the integral sign, note that
$$\int_{0}^{1} x^t (\ln(x))^n \, dx=\frac{d^n}{dt^n}\left( \int_0^1 x^t dx\right)=\frac{d^n}{dt^n} \left((t+1)^{-1}\right).$$
$$
\int_{0}^{1}\! \left( x\ln  \left( x \right)  \right) ^{n}\,{\rm d}x={
\frac { \left( -1 \right) ^{n}n!}{ \left( n+1 \right) ^{n+1}}}
$$