Sinx / x dx Solution I = integration ( 0 to infinty ) e -sx cos(x) dx J = integration ( 0 to infinty ) e-sx sin(x)dx use integration by parts - I = e-sx ( sin (x )) + s J J = - e-sx ( cos(x ))- s I by solving the above equations - J = 1 / ( s2 +1 ) integration ( oto infinity ) integration ( 0 to infinity ) ( e- sxsin(x)ds )dx = integration ( 0 to infinty ) sin(x) / x dx integration ( 0 to infinty ) 1 / s2 +1 ds = integration ( 0 to infinty ) ( sin(x) / x ) dx pi / 2 = integration ( 0 to infinty ) sin(x) / x dx.