Prove the relation $\frac{1}{x}$=$\int^\infty_0$ $e^{-xt}$ dt, for $x>0$. Use it to prove $\int^\infty_0$ $\frac{\sin(x)}{x}$ dx = $\frac{\pi}{2}$
Evaluating the integral $\int_0^\infty \frac{\sin x} x \,\mathrm dx = \frac \pi 2$?