How do you evaluate the following integral analytically (no graphs, or calculators)?

Question

anonymous · Answer

$$\int\limits_{0}^{5}\left| (x^2-5x+4) \right| dx$$

anonymous · Answer

Thanks!  How did you get those 3 parts (this is kinda dumb, but sorry).

anonymous · Answer

when you put x=1   the absolute value is positive 

when you put x=2 and three  absolute value is negative  (thats why i put - sign in front of second integral )

for  x=4 and 5  absolute value is again positive .

anonymous · Answer

Isn't it 0 @ 1?

anonymous · Answer

yes it is because at x=0  still abs (x^-5x+4)  =(0-0+4)=4  positive .

anonymous · Answer

Should the second integral be from 1 to 4 instead of 1 to 3, and the third integral be from 4 to 5?

anonymous · Answer

at x=1   yes zero .

anonymous · Answer

so the second integral should be from 1 to 4, and the 3rd one 4 to 5?

anonymous · Answer

sorry  i wrote it wrong . you are right 
first from 0..1  second from 1..4  and third from 4..5

anonymous · Answer

$$\Large  \int\limits_{0}^{1}  (x^2-5x+4)dx-\int\limits_{1}^{4}  (x^2-5x+4)dx+\int\limits_{4}^{5}(x^2-5x+4)dx$$