Question

D/dxtle=" \int\limits^0_x {cos(2 \pi u)} \, du" alt=" \int\limits^0_x {cos(2 \pi u)} \, du" align="absmiddle" class="latex-formula"> is

Answer 1

$\displaystyle\frac{\mathrm d}{\mathrm dx}\int_x^0\cos2\pi u\,\mathrm du=-\frac{\mathrm d}{\mathrm dx}\int_0^x\cos2\pi u\,\mathrm du=-\cos2\pi x$

by the fundamental theorem of calculus.