Question

como calcular essa integral?

Answer 1

Olá


$\displaystyle\mathsf{ \int\limits^{ \frac{\pi}{3} }_0 {(3+cos(3x))} \, dx }$



Divide a integral em uma soma de integrais


$\displaystyle\mathsf{ \int\limits^{ \frac{\pi}{3} }_0 {3} \, dx ~+~\int\limits^{ \frac{\pi}{3} }_0 {cos(3x)} \, dx}$


A primeira integral é simples. ∫3dx = 3x + c

A segunda integral, há uma propriedade que diz o seguinte:


$\displaystyle\mathsf{\int cos(\delta x)dx~=~ \frac{sen(\delta x)}{\delta}+C\qquad\qquad\qquad\delta \in\Re }\\\\\\\mathsf{\int sen(\delta x)dx~=~ -\frac{cos(\delta x)}{\delta}+C\qquad\qquad\qquad\delta \in\Re }$


usando dessa propriedade


$\displaystyle\mathsf{ \int\limits^{ \frac{\pi}{3} }_0 {3} \, dx ~+~\int\limits^{ \frac{\pi}{3} }_0 {cos(3x)} \, dx}\\\\\\\\\mathsf{\left(3x+ \frac{sen(3x)}{3} \right)\bigg|^{ \frac{\pi}{3} }_{0}}\\\\\\\\\mathsf{\left(3\cdot \frac{\pi}{3} ~+~ \frac{sen( \frac{\pi}{3} )}{3} \right)~-~\left(3\cdot 0~+~ \frac{sen(0 )}{3} \right)}$

$\displaystyle \mathsf{\left(\diagup\!\!\!\!3\cdot \frac{\pi}{\diagup\!\!\!\!3} ~+~ \underbrace{\frac{sen( \frac{\pi}{3} )}{3}}_{=0} \right)~-~\left(\underbrace{3\cdot 0~+~ \frac{sen(0 )}{3}}_{=0} \right)}\\\\\\\\\mathsf{( \pi +0)~-~(0+0)}\\\\\\\boxed{\mathsf{\pi}}$