Can someone help me with this question.
Evaluate the integral fro 0 to inifinity of sinx over sinh x ?
Thanks
$\displaystyle \int_0^{\infty} \frac{\sin x}{\sinh x}dx$.
Instead we will do,
$\displaystyle \int_{-\infty}^{\infty} \frac{\sin x}{\sinh x} dx$
Because the function is even.
Now define the complex-valued function,
$\displaystyle f(z) = \frac{\sin z}{\sinh z}$.
The singularities of this function occur when $\displaystyle \sinh z = 0$. Since there are infinitely many such values we cannot use the semi-circular contour. Let us use a rectangular contour: $\displaystyle -R,+R,-R+\frac{3}{2}\pi i,R+\frac{3}{2}\pi i$.
Can you finish?