Obtain for n > 0 a relation of the form $I(m, n) = kI(m, n − 1)$

Question

The function $I(m, n)$, where $m ≥ 0$ and $n ≥ 0$ are integers, is defined by

$$I(m, n) = \int_{0}^{1} x^m(-\ln x)^n dx$$

Obtain for $n > 0$ a relation of the form $I(m, n) = kI(m, n − 1)$, where $k$ is to be found. Hence obtain an explicit formula for $I(m, n)$, and show that $I(5, 4) = \dfrac{1}{324}$

Answer 1

This is just a problem of integration by parts $$u=(-\log (x))^n \implies du=-\frac{n (-\log (x))^{n-1}}{x}$$ $$dv=x^m dx \implies v=\frac{x^{m+1}}{m+1}$$ $$I(m, n) = \int x^m(-\ln x)^n dx=\frac{x^{m+1} (-\log (x))^n}{m+1}+\frac n {m+1}\int{ x^m (-\log (x))^{n-1}}\,dx$$