Question

Calcule a primitiva (integral) da função abaixo.

Answer 1

✅ Após resolver os cálculos, concluímos que a primitiva da referida função é:

$\Large\displaystyle\text{ \begin{gathered}\boxed{\boxed{\:\:\:\bf \int \Bigg(\sqrt{x} + \frac{1}{\sqrt{x}}\Bigg)dx= \frac{2\sqrt{x^{3}}}{3} + 2\sqrt{x} + c\:\:\:}}\end{gathered} }$

Seja a função:

               $\Large\displaystyle\text{ \begin{gathered}\tt f(x) = \sqrt{x} + \frac{1}{\sqrt{x}}\end{gathered} }$

Calculando a primitiva da referida função, temos:  

$\Large\displaystyle\text{ \begin{gathered}\tt \int f(x)dx = \int\Bigg(\sqrt{x} + \frac{1}{\sqrt{x}}\Bigg)dx\end{gathered} }$

                     $\Large\displaystyle\text{ \begin{gathered}\tt = \int\sqrt{x}dx + \int\frac{1}{\sqrt{x}}dx\end{gathered} }$

                     $\Large\displaystyle\text{ \begin{gathered}\tt = \int x^{\frac{1}{2}}dx + \int \frac{1}{x^{\frac{1}{2}}}dx\end{gathered} }$

                     $\Large\displaystyle\text{ \begin{gathered}\tt = \int x^{\frac{1}{2}}dx + \int x^{-\frac{1}{2}}dx\end{gathered} }$

                     $\Large\displaystyle\text{ \begin{gathered}\tt = \frac{x^{\frac{1}{2} + 1}}{\frac{1}{2} + 1} + \frac{x^{-\frac{1}{2} + 1}}{-\frac{1}{2} + 1} + c\end{gathered} }$

                     $\Large\displaystyle\text{ \begin{gathered}\tt = \frac{x^{\frac{3}{2}}}{\frac{3}{2}} + \frac{x^{\frac{1}{2}}}{\frac{1}{2}} + c\end{gathered} }$

                      $\Large\displaystyle\text{ \begin{gathered}\tt = x^{\frac{3}{2}}\cdot\frac{2}{3} + x^{\frac{1}{2}}\cdot\frac{2}{1} + c\end{gathered} }$

                      $\Large\displaystyle\text{ \begin{gathered}\tt = \frac{2x^{\frac{3}{2}}}{3} + 2x^{\frac{1}{2}} + c\end{gathered} }$

                      $\Large\displaystyle\text{ \begin{gathered}\tt = \frac{2\sqrt{x^{3}}}{3} + 2\sqrt{x} + c\end{gathered} }$

Portanto, a primitiva da função é:

       $\Large\displaystyle\text{ \begin{gathered}\tt \int \Bigg(\sqrt{x} + \frac{1}{\sqrt{x}}\Bigg)dx= \frac{2\sqrt{x^{3}}}{3} + 2\sqrt{x} + c\end{gathered} }$

Saiba mais:

1. https://brainly.com.br/tarefa/241726
2. https://brainly.com.br/tarefa/46169101