Question: How does$$T=-\int\limits_{\tfrac {\pi}2}^{-\tfrac {\pi}2}dx\,\frac {\log\left(\tfrac {a^2}4\cos^2x\right)}{\sqrt{\tfrac {a^2}4\cos^2x}}\left(\frac a2\cos x\right)=4\int\limits_0^{\tfrac {\pi}2}dx\,\log\left(\frac a2\cos x\right)\tag1$$
I was evaluating an integral, namely$$I=\int\limits_0^adx\,\frac {\log x}{\sqrt{ax-x^2}}$$And I'm having trouble seeing how to get from the left-hand side to the right-hand side. Doesn't the denominator and numerator cancel to leave you with$$T=-\int\limits_{\tfrac {\pi}2}^{-\tfrac {\pi}2}dx\,\log\left(\frac {a^2}4\cos^2x\right)=-2\int\limits_{\tfrac {\pi}2}^{-\tfrac {\pi}2}dx\,\log\left(\frac a2\cos x\right)$$But I don't see how that can be substituted to get the second expression of $(1)$. What kind of substitution should be made to make the transformation from where I left off to $(1)$?