\begin{align} &\int_{a}^{b}f(x)dx = \int_{a}^{b}f(a+b-x)dx\\ =&\frac{1}{2}\int_{a}^{b}[f(x)+f(a+b-x)]dx \end{align}

\begin{align} &\int_{0}^{\frac{\pi}{2}}lnsinxdx\\ =&\int_{0}^{\frac{\pi}{2}}lnsin(\frac{\pi}{2}-x)dx\\ =&\int_{0}^{\frac{\pi}{2}}lncosxdx\\ =&\frac{1}{2}\int_{0}^{\frac{\pi}{2}}ln(\frac{sin2x}{2})dx\\ =&\frac{1}{2}\int_{0}^{\frac{\pi}{2}}lnsin2xdx-\frac{\pi}{4}ln2 \end{align}