TITLE: How do I evaluate the following definite integral: $\int^1_0\frac{x^2\ln x}{\sqrt{1-x^2}}dx$?
QUESTION [5 upvotes]: $$\int^1_0\frac{x^2\ln x}{\sqrt{1-x^2}}dx$$
I tried using by parts but it didn't help. Also none of the properties of definite integral helps.

REPLY [3 votes]: Rewrite the integral as$$I(a)=\lim\limits_{a\to0}\frac {\partial}{\partial a}\int\limits_0^1dx\,\frac {x^{a+2}}{\sqrt{1-x^2}}$$And make the substitution $u=x^2$. You should get the resulting integral in terms of the beta function.