int_2^8 |5-x| dx = int_2^5 (5-x)dx + int_5^8 (x-5)dx
= [ 5x - x^2/2 + C1 ]_2^5 + [ x^2/2 - 5x + C2 ]_5^8
= 12.5 + C1 - 8 - C1 - 8 + C2 + 12.5 - C2
= 9
"In the first step we just apply the definition of |...| :"
|x| = {(-x, ", "x<=0),(x, ", "x >= 0):}
"So"
|5 - x| = { (x - 5, ", "5-x <= 0), (5 - x, ", "5-x >= 0):}
= {(x - 5, ", "x >= 5), (5 - x, ", "x <= 5):}
"So the limit case x=5 splits the integration interval up in two"
"parts : [2 ,5] and [5, 8]."