Question

Calcule a integral definida:
$\int\limits^1_0 {(sen \pi x.cos \pi x)} \, dx$

Answer 1

$\displaystyle I=\int\limits_{0}^1\sin \pi x \cdot \cos \pi x\; dx \\ \\ I=\frac{1}{\pi}\int\limits_{0}^1\sin \pi x \; d(\sin \pi x) \\ \\ I=\frac{1}{\pi}\left.\left(\frac{\sin^2 \pi x}{2}\right)\right|_0^1\\ \\ I=\frac{1}{\pi}\left(\frac{\sin^2 \pi }{2}-\frac{\sin^2 0 }{2}\right)\\ \\ \\ \boxed{I=0}$