Triple integral by José Sacramento

$\Large \int_{0}^{\pi}\int_{\sqrt{z}}^{\sqrt{\pi}}\int_{0}^{x} sin(xy) dydxdz$ $= \Large \int_{0}^{\pi}\int_{\sqrt{z}}^{\sqrt{\pi}}\frac{1-cos(x^{2})}{x} dxdz$ Now changing the order of integration $= \Large \int_{0}^{\sqrt{\pi}}\int_{0}^{x^{2}}\frac{1-cos(x^{2})}{x} dzdx$ $= \Large \int_{0}^{\sqrt{\pi}}{x(1-cos(x^{2}))}dx$ $= \Large \frac{\pi}{2} + \int_{0}^{\pi}\frac{cos(t)}{2}dt$ (Here I have used the substitution $x^{2} = t$ .)

(Also $\int_{0}^{\pi}cos(x)dx = 0$ ) $= \frac{\pi}{2}$