Question

O valor da integral ∫_0^1[ ∫_(x^2)^x dy] dx é:

Answer 1

Resolve-se usando o Teorema de Fubini (integrais iteradas):

$\displaystyle\int_0^1\left[\int_{x^2}^x dy\right]dx\\\\\\ =\int_0^1y\big|_{x^2}^x\,dx\\\\\\ =\int_0^1(x-x^2)\,dx\\\\\\ =\left(\frac{x^2}{2}-\frac{x^3}{3}\right)\bigg|_0^1\\\\\\ =\frac{1^2}{2}-\frac{1^3}{3}\\\\\\ =\frac{1}{2}-\frac{1}{3}$

$=\dfrac{3}{6}-\dfrac{2}{6}\\\\\\ =\dfrac{1}{6}\quad\quad\checkmark$


Dúvidas? Comente.


Bons estudos! :-)