a quite elementary proof of the fact, that the integral of sin x/x between 0 and infinity is equal to pi/2 runs as follows:
1.) Show, that the integral (conditionally) converges.
2.) Because of (sin (2n+1) x)/sin x = 1 + 2cos2x + 2 cos 4x + ....+ 2 cos 2nx (n a natural number) follows, that the integral of (sin (2n+1) x)/sin x between 0 and pi/2 is equal to pi/2 for all natural n.
3.) Show, that the function f(x) = 1/x - 1/sin x has a continuous derivative for all x in [0; pi/2].
4.) Show by partial integration, that the integral of (sin (2n+1)x)(1/x - 1/sin x) between 0 and pi/2 tends to 0 if n tends to infinity.
Then we are done, because the integral of sin x/x between 0 and infinity is equal to the limit of the integral of sin (2n+1) x/x between 0 and pi/2 as n tends to infinity.
In principle, the idea of this proof is implicitly contained in Dirichlet's articles on the convergence of Fourier series. The earliest complete version of this proof, which I can find, however, is in Courant's "Differential- und Integralrechnung", Vol. I, 2nd(!) ed., 1930, pp. 371/72.
Does anybody know an earlier source for this particular proof?