Question

Salut, am nevoie de ajutor la ex. 3

Answer 1

$\displaystyle I =\int_{0}^2 \Big|x^2+\big|x^2-x\big|-1\Big|\, dx\\ \\ = \int_{0}^2 \Big|(x-1)(x+1)+\big|x(x-1)\big|\Big|\, dx \\ \\ \text{Daca }x \in[0,1]:\\f(x) =|(x-1)(x+1)-x(x-1)| =\\ = |(x-1)(x+1-x)| = |(x-1)| = 1-x\\ \\ \text{Daca }x\in[1,2]:\\\ f(x) = (x-1)(x+1)+x(x-1) = (x-1)(x+1+x)=\\ =(x-1)(2x+1)\\ \\ I = \int_{0}^1 (1-x)\, dx+\int_{1}^2(2x^2-x-1)\, dx = \boxed{\dfrac{8}{3}}$