Question

O valor da integral ∫π,0 senx cosx dx é?

Answer 1

$\displaystyle \int\limits^0_\pi {\sin(x)\cdot\cos(x)} \, dx =-\int\limits^{\pi}_{0}\sin(x)\cdot\cos(x) \, dx\\\\ u=\sin(x)\\\\du=u'\cdot dx\implies u'=cos(x)\\\\-\int\limits^{\pi}_{0}u\, du=-\int\limits^{\pi}_{0}u\cdot u'\, dx\implies\\\\ -\int\limits_{0}^{\pi}u\ du=-\frac{u^{1+1}}{1+1}\big|\limits^{\pi}_{0}=-\frac{u^2}{2}\big|\limits^{\pi}_{0}\implies u=\sin(x)\implies -\frac{\sin^2(x)}{2}\big|\limits_{0}^{\pi}$$\displaystyle \\\\ -\frac{\sin^2(\pi)}{2}-(-\frac{\sin^2(0)}{2})\implies -\frac{\sin^2(\pi)}{2}+\frac{\sin^2(0)}{2}=\frac{0}{2}-\frac{0}{2}=\boxed{0}$