Question

Primitiva da função f (x) = √x

Answer 1

$\mathtt{f(x)=\sqrt{x}}\\\\ \mathtt{f(x)=x^{\frac{1}{2}}}$


Regra para primitiva de potências:

$\displaystyle\mathtt{\int x^n\,dx=}\left\{ \begin{array}{ll} \mathtt{\dfrac{x^{n+1}}{n+1}}&\texttt{ se }\mathtt{n\ne -1}\\\\ \mathtt{\ell n|x|}&\texttt{ se }\mathtt{n=-1} \end{array} \right.$

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Dessa forma, temos que

$\displaystyle\mathtt{\int f(x)\,dx}\\\\\\ =\mathtt{\int x^{\frac{1}{2}}\,dx}\\\\\\ =\mathtt{\frac{x^{\frac{1}{2}+1}}{\frac{1}{2}+1}+C}\\\\\\ =\mathtt{\frac{x^{\frac{3}{2}}}{\frac{3}{2}}+C}\\\\\\ =\mathtt{\frac{2}{3}\,x^{\frac{3}{2}}+C}\\\\\\ =\boxed{\begin{array}{c}\mathtt{\dfrac{2}{3}\sqrt{x^3}+C} \end{array}}$


Dúvidas? Comente.


Bons estudos! :-)