Question

Qual a antiderivada da função f(x) = 3x² + √x + 5.

Answer 1

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

$\Large\displaystyle\text{ \begin{gathered}\boxed{\boxed{\:\:\:\bf F(x) = x^{3} + \frac{2\sqrt{x^{3}}}{3} + 5x + c\:\:\:}}\end{gathered} }$  

Seja a função:

             $\Large\displaystyle\begin{gathered}\tt f(x) = 3x^{2} + \sqrt{x} + 5\end{gathered}$

Calculando a antiderivada de "f(x)", temos:

      $\Large\displaystyle\begin{gathered}\tt F(x) = \int f(x)\,dx\end{gathered}$

                 $\Large\displaystyle\begin{gathered}\tt = \int (3x^{2} + \sqrt{x} + 5)\,dx\end{gathered}$

                 $\Large\displaystyle\begin{gathered}\tt = \int 3x^{2}\,dx + \int \sqrt{x} + \int 5\,dx\end{gathered}$

                 $\Large\displaystyle\text{ \begin{gathered}\tt = 3\cdot\int x^{2}\,dx + \int x^{\frac{1}{2} } + 5\cdot \int dx\end{gathered} }$

                  $\Large\displaystyle\text{ \begin{gathered}\tt = 3\cdot\frac{x^{2 + 1}}{2 + 1} + \frac{x^{\frac{1}{2} + 1}}{\frac{1}{2} + 1} + 5x + c\end{gathered} }$

                  $\Large\displaystyle\text{ \begin{gathered}\tt = \frac{3}{3}x^{3} + \frac{x^{\frac{3}{2}}}{\frac{3}{2}} + 5x + c\end{gathered} }$

                   $\Large\displaystyle\text{ \begin{gathered}\tt = x^{3} + \frac{2}{3}x^{\frac{3}{2}} + 5x + c\end{gathered} }$

                   $\Large\displaystyle\text{ \begin{gathered}\tt = x^{3} + \frac{2\sqrt{x^{3}}}{3} + 5x + c\end{gathered} }$

Portanto, a antiderivada procurada é:

    $\Large\displaystyle\text{ \begin{gathered}\tt F(x) = x^{3} + \frac{2\sqrt{x^{3}}}{3} + 5x + c\end{gathered} }$

         

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