Question

Calcule a integral definida:

Answer 1

$\displaystyle\int\limits_{\pi/3}^{\pi}{\cos\left(\frac{x}{2}\right)dx}$


Fazendo a seguinte substituição:

$u=\dfrac{x}{2}~\Rightarrow~x=2u~\Rightarrow~dx=2\,du$


Mudando os extremos de integração:

$\text{Quando }x=\dfrac{\pi}{3}~\Rightarrow~u=\dfrac{\pi}{6}\\ \\ \\ \text{Quando }x=\pi~\Rightarrow~u=\dfrac{\pi}{2}$


Substituindo a integral fica

$\displaystyle\int\limits_{\pi/6}^{\pi/2}{\cos(u)\cdot 2\,du}\\ \\ \\ =2\int\limits_{\pi/6}^{\pi/2}{\cos(u)\,du}\\ \\ \\ =2\cdot \left.\mathrm{sen}(u)\right|_{\pi/6}^{\pi/2}\\ \\ \\ =2\cdot \left(\mathrm{sen}\,\frac{\pi}{2}-\mathrm{sen}\,\frac{\pi}{6} \right )\\ \\ \\ =2\cdot \left(1-\frac{1}{2} \right )\\ \\ \\ =2\cdot \cdot \frac{1}{2}\\ \\ \\ =1$