Question

O valor resultante do cálculo da integral definida ∫0^1 (1 - ∛x) dx corresponde a:

Answer 1

Calcular a integral definida:

     $\displaystyle\int_0^1 (1-\,^3\!\!\!\sqrt{x})\,dx\\\\\\ \int_0^1 (1-x^{1/3})\,dx$


Aplique a regra para obter primitivas de potências:

     •  $\displaystyle\int x^n\,dx=\dfrac{x^{n+1}}{n+1}+C\qquad\quad\mathsf{para~}n\ne-1$


Aplicando o Teorema Fundamental do Cálculo, a integral fica

     $=\bigg(x-\dfrac{x^{(1/3)+1}}{\frac{1}{3}+1}\bigg)\bigg|_0^1\\\\\\ =\bigg(x-\dfrac{x^{4/3}}{\frac{4}{3}}\bigg)\bigg|_0^1\\\\\\ =\bigg(x-\dfrac{3}{4}\,x^{4/3}\bigg)\bigg|_0^1\\\\\\ =\bigg(1-\dfrac{3}{4}\cdot 1^{4/3}\bigg)-\bigg(0-\dfrac{3}{4}\cdot 0^{4/3}\bigg)\\\\\\ =1-\dfrac{3}{4}\\\\\\ =\dfrac{4}{4}-\dfrac{3}{4}$

     $=\dfrac{1}{4}\quad\longleftarrow\quad \mathsf{resposta.}$


Dúvidas? Comente.


Bons estudos! :-)