SEEMOUS 2019 Problem 4

(a) Let $n$ is a positive integer. Calculate $\int_0^1 x^{n-1}\ln x dx$ .

(b) Calculate $\sum_{n=0}^{\infty}(-1)^n\left(\frac{1}{(n+1)^2}-\frac{1}{(n+2)^2}+\frac{1}{(n+3)^2}-\dots \right).$