Question

O resultado da integral dx 3√ 3x+7dx no limite de 2 a -2 é:

Answer 1

$\displaystyle\mathsf{\int\limits_{-2}^{2}3\sqrt{3x+7}\,dx}\\\\\\ =\mathsf{\int\limits_{-2}^{2}\sqrt{3x+7}\cdot 3\,dx}~~~~~~\mathbf{(i)}$


Fazendo a seguinte mudança de variável:

$\mathsf{3x+7=u~~\Rightarrow~~3\,dx=du}$


Mudando os extremos de integração:

$\mathsf{Quando~~x=-2~~\Rightarrow~~u=1}\\\\ \mathsf{Quando~~x=2~~\Rightarrow~~u=13}$


Substituindo em $\mathbf{(i)},$ a integral fica

$\displaystyle\mathsf{=\int\limits_{1}^{13}\sqrt{u}\,du}\\\\\\ \mathsf{=\int\limits_{1}^{13}u^{1/2}\,du}\\\\\\ \mathsf{=\left.\left(\dfrac{u^{(1/2)+1}}{\frac{1}{2}+1} \right )\right|_{1}^{13}}\\\\\\ \mathsf{=\left.\left(\dfrac{u^{3/2}}{\frac{3}{2}} \right )\right|_{1}^{13}}\\\\\\ \mathsf{=\left.\left(\dfrac{2}{3}\,u^{3/2} \right )\right|_{1}^{13}}\\\\\\ \mathsf{=\dfrac{2}{3}\cdot (13^{3/2}-1^{3/2})}\\\\\\ \mathsf{=\dfrac{2}{3}\left(13\sqrt{13}-1\right)}$