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Question
By using the properties of the definite integral, evaluate the integral:
`int_(-5)^5 | x + 2| dx`
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Solution
Let `I = int_-5^5 abs (x + 2) dx`
Define,
`abs (x + 2) = {(-(x + 2), if x + 2 < 0, or x< - 2),(x + 2, if x +2 >= 0, or x >=-2):}`
∵ `I = - int_-5^-2 (x + 2) dx + int_-2^5 (x + 2) dx`
`= -[(x + 2)^2/2]_-5^-2 + [(x + 2)^2/2]_-2^5`
`= [((-2 + 2)^2/2 - (-5 + 2)^2/2)] + [(5 + 2)^2/2 - (-2 + 2)^2/2]`
`= -1/2 [-9] + 1/2 [49 - 0]`
`= 9/2 + 49/2`
`= 58/2`
= 29
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