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Question
By using the properties of the definite integral, evaluate the integral:
`int_2^8 |x - 5| dx`
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Solution
`int_2^8 abs (x - 5) dx`
Define,
`abs(x - 5) = {(-(x - 5), if x - 5 < 0, or x< 5),(x - 5, if x - 5 >= 0, or x >=5):}`
`= int_2^5 abs (x - 5) dx + int_2^8 abs (x - 5) dx`
`= - int_2^5 (x - 5) dx + int_2^8 (x - 5) dx`
`= - [x^2/2 - 5x]_2^5 + [x^2/2 - 5x]_5^8`
`= - [25/2 - 25 - 4/2 + 10]`
`= [64/2 - 4 - 25/2 + 25]`
`= - [(-9)/2] + [9/2]`
`= 9/2 + 9/2`
= 9
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