Question

(30 PONTOS) Calcule a integral definida:
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$\displaystyle\int\limits_{0}^{2\pi/3}{\mathrm{sen}\!\left(\dfrac{\pi}{3}-\left|x-\dfrac{\pi}{3} \right| \right )dx}$
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Sugestão: Utilize a simetria da função $f(x)=\mathrm{sen}\!\left(\dfrac{\pi}{3}-\left|x-\dfrac{\pi}{3}\right| \right ).$
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Answer 1

$\displaystyle I=\int\limits_0^{2\pi/3}\sin\left(\frac{\pi}{3}-\left|x-\frac{\pi}{3} \right| \right )dx\\ \\ \\ I=\int\limits_0^{\pi/3}\sin\left(\frac{\pi}{3}-\left|x-\frac{\pi}{3} \right| \right )dx+\int\limits_{\pi/3}^{2\pi/3}\sin\left(\frac{\pi}{3}-\left|x-\frac{\pi}{3} \right| \right )dx\\ \\ \\ I=\int\limits_0^{\pi/3}\sin x\,dx+\int\limits_{\pi/3}^{2\pi/3}\sin\left(\frac{2\pi}{3}-x \right )dx$


$\displaystyle I=\left.-\cos x\right|_{0}^{\pi/3}}+\left.\cos\left(\frac{2\pi}{3}-x \right)\right|_{\pi/3}^{2\pi/3}\\ \\ \\ I=2\left(\cos 0 - \cos \dfrac{\pi}{3}\right)\\ \\ \\ I=2\left(1-\dfrac{1}{2}\right)\\ \\ \\ \boxed{I=1}$