Question

A antiderivada mais geral da função g(x)=x^2sen x é:

Answer 1

antiderivada é a integral.

Temos :

$\displaystyle \text{g(x)}=\text x^2+\text{sen(x)} \\\\\ \underline{\text{integrando}}: \\\\\\ \text{G(x)}=\int [ \ \text x^2+\text{sen(x)}\ ] \text{dx} \\\\\\ \text{G(x)}=\int \text x^2.\text{dx} +\int \text{sen(\text x)}\text{dx}\\\\\\ \text{G(x)}= \frac{\text x^3}{3} + [-\text{cos(x)}]+\text C \\\\\\ \huge\boxed{\text{\ G(x)}= \frac{\text x^3}{3}-\text{cos(x)}+\text C\ }\checkmark \\\\\\ \text{(letra c)}$

Answer 2

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

$\Large\displaystyle\text{ \begin{gathered}\boxed{\boxed{\:\:\:\bf G(x) = \frac{1}{3}x^{3} - \cos(x) + c\:\:\:}}\end{gathered} }$

Seja a função:

         $\Large\displaystyle\begin{gathered}\tt g(x) = x^{2} + \sin(x)\end{gathered}$

Calculando a primitiva "G(x)" da função "g(x)", temos:

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

                     $\Large\displaystyle\begin{gathered}\tt = \int \left[x^{2} + \sin(x)\right]\,dx\end{gathered}$

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

                     $\Large\displaystyle\text{ \begin{gathered}\tt = \frac{x^{2 + 1}}{2 + 1} + \left[-\cos(x)\right] + c\end{gathered} }$

                     $\Large\displaystyle\begin{gathered}\tt = \frac{1}{3}x^{3} - \cos(x) + c\end{gathered}$

✅ Portanto, a primitiva procurada é:

           $\Large\displaystyle\begin{gathered}\tt G(x) = \frac{1}{3}x^{3} - \cos(x) + c\end{gathered}$

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Solução gráfica (figura):