Question: Evaluate$$I=\int\limits_0^{1/2}dx\,\frac {\log x}{\sqrt{1-x^2}}$$
I've given it the best I could. The first thing I did was use differentiation under the integral sign, but I wasn't sure where to continue after that.
The next thing I did was try substituting $x\mapsto 2x$ to get the limits in terms of zero and one and transform $I$ into the beta function (no luck because the denominator is $1-4x^2$).
Then I tried integration by parts. It seemed promising at first because I got$$I=-\frac {\pi}6-\int\limits_0^{1/2}dx\,\frac {\arcsin x}{x}$$But I'm not sure how to evaluate the second integral. I even tried using a Laplace Transform, but I'm again not sure how to evaluate the integral$$\mathcal{L}(f(t))=\int\limits_0^{\infty}dt\, e^{-st}\arcsin t$$
Can anybody give me a few hints as to where to begin and what to do? I'm out of ideas.