Question

$\int\limits^3_1 |x+1|+|x-2|} \, dx$

Answer 1

$\it \int\limits^3_1 (|x+1|+|x-2|) dx$

Explicităm modulele:


[tex]\it |x+1| = x+1 ,\ \ \forall x\in [1, \ 3 ] \\\;\\ |x-2| = -x +2 ,\ \ \forall x\in [1, \ 2 ] \\\;\\ |x-2| = x -2 ,\ \ \forall x\in [2, \ 3 ] [/tex]

Integrala devine:

[tex]\it \int\limits^2_1 (x+1-x+2) dx + \int^3_2 (x+1+x-2) dx = \int\limits^2_1 3 dx+ \int^3_2 (2x-1) dx= \\\;\\ \\\;\\ =3x\Big|^2_1 + (x^2-x)\Big|^3_2 = 6-3+6-2 = 7[/tex]