Partial Derivatives & the Gradient · Superchallenge problem

For $x > 0$ and real $y$ let $F(x,y) = \displaystyle\int_0^{\infty} e^{-xt}\,\frac{\sin(yt)}{t}\,dt$. Compute $F_y$ by differentiating under the integral sign, and use it to find $F\!\left(1, \sqrt{3}\right)$.