3k views

2k views

### Can $\int_0^{\pi/2} \ln ( \sin(x)) \; dx$ be evaluated with “complex method”?

Can the following integral be evaluated using complex method by substituting $\sin(x) = {e^{ix}-e^{-ix} \over 2i}$? $$I=\int_0^{\pi/2} \ln ( \sin(x)) \; dx = - {\pi \ln(2) \over 2}$$
73 views

### Big trouble in evaluating an Improper integral in complex analysis [duplicate]

I am having trouble in evaluating $\int_{0}^\infty \left(\frac{log(x^2+1)}{x^2+1}\right)dx$ I proceeded by taking $\int_{C}\left(\frac{log(z+i)}{z^2+1}\right)dz$ I don't know how to draw contour here ...
### Evaluate $\int_0^\infty \frac{\sqrt{x}}{x^2+1}\log\left(\frac{x+1}{2\sqrt{x}}\right)\;dx$
Prove that $$\int_0^\infty \frac{\sqrt{x}}{x^2+1}\log\left(\frac{x+1}{2\sqrt{x}}\right)\;dx=\frac{\pi\sqrt{2}}{2}\log\left(1+\frac{\sqrt{2}}{2}\right).$$ I managed to prove this result with some ...