Question

qual é a integral de raiz de 3 - 2x dx

Answer 1

Calcular a integral indefinida:

     $\mathsf{\displaystyle\int\sqrt{3-2x}\,dx}\\\\\\ \mathsf{=\displaystyle\int\sqrt{3-2x}\cdot \Big(\!-\!\frac{1}{2}\Big)\cdot (-2)\,dx}\\\\\\ \mathsf{=\displaystyle -\,\frac{1}{2}\int\sqrt{3-2x}\cdot (-2)\,dx}$


Faça a seguinte substituição:

     $\mathsf{u=3-2x\quad\Longrightarrow\quad du=-2\,dx}$


e a integral fica

     $\mathsf{=\displaystyle -\,\frac{1}{2}\int\sqrt{u}\,du}\\\\\\ \mathsf{=\displaystyle -\,\frac{1}{2}\int u^{1/2}\,du}$


Aplique a regra para encontrar primitivas de potências:

     •  $\mathsf{\displaystyle\int x^n\,dx=\dfrac{x^{n+1}}{n+1}+C,\qquad para~n\ne -1}$


e a integral fica

     $\mathsf{=-\,\dfrac{1}{2}\cdot \dfrac{u^{\frac{1}{2}+1}}{\frac{1}{2}+1}+C}\\\\\\ \mathsf{=-\,\dfrac{1}{2}\cdot \dfrac{u^{3/2}}{\frac{3}{2}}+C}\\\\\\ \mathsf{=-\,\dfrac{1}{\diagup\!\!\!\! 2}\cdot \dfrac{\diagup\!\!\!\! 2}{3}\,u^{3/2}+C}\\\\\\ \mathsf{=-\,\dfrac{1}{3}\,u^{3/2}+C}$


Substitua de volta para a variável x:

     $\mathsf{=-\,\dfrac{1}{3}\,(3-2x)^{3/2}+C\quad\longleftarrow\quad esta~\acute{e}~a~resposta.}$


Dúvidas? Comente.


Bons estudos! :-)