Question

Salut, am nevoie de ajutor la ex. 1025( explicatii pas cu pas )

Answer 1

$\displaystyle I = \int_{-2}^22^{[x]}\, dx \\ \\ \\ 2^{[x]} = \begin{cases} 2^{-2},\ x\in(-2,-1) \\ 2^{-1},\ x\in(-1,0)\\2^0,\ x\in(0,1)\\2^1,\ x\in(1,2)\end{cases} \\ \\ \\ I = \int_{-2}^{-1}\dfrac{1}{4}\, dx+ \int_{-1}^{0}\dfrac{1}{2}\, dx + \int_{0}^{1}1\, dx+ \int_{1}^{2}2\, dx = \\ \\ \\ =\dfrac{-1}{4}+\dfrac{2}{4}+\dfrac{1}{2}+1+4-2 = \\ \\ = \dfrac{1+2+12}{4} = \boxed{\dfrac{15}{4}}$