Question

calcule a função primitiva de 1/x√x

Answer 1

Boa noite

f(x) = 1/(x√x) 

primitiva

F(x) = -2/√x + C 

Answer 2

✅ Após resolver os cálculos, concluímos que a primitiva - antiderivada ou integral indefinida - da função dada é:

 $\Large\displaystyle\text{ \begin{gathered}\boxed{\boxed{\:\:\:\bf \int \frac{1}{x\sqrt{x}}\,dx= -\frac{2}{\sqrt{x}} + c\:\:\:}}\end{gathered} }$

Seja a função dada:

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

Para encontrar a primitiva desta função, devemos resolver a seguinte integral indefinida:

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

Então, temos:

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

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

                           $\Large\displaystyle\text{ \begin{gathered}\tt = \int \frac{1}{x^{\frac{3}{2}}}\,dx\end{gathered} }$

                           $\Large\displaystyle\text{ \begin{gathered}\tt = \int x^{-\frac{3}{2}}\,dx\end{gathered} }$

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

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

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

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

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

✅ Portanto, a primitiva procurada é:

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

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Veja a solução gráfica representada na figura: