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Evaluate the definite integrals `int1 4[x-1|+|x-2|+|x-3|]dx`

Text Solution

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`int_(1)^(4) {(|x-1|+|x-2|+|x-3|)}dx`
`= int_(2)^(3) (|x-1|+|x-2|+|x-3|)`
`+int_(2)^(3) {|x-1|+|x-2|+|x-3|}dx`
`+int_(3)^(4) {|x-1|+|x-2|+|x-3|}dx`
`=int_(1)^(2) {x-1-(x-2)-(x-3)}dx`
`+int_(2)^(3) {x-1+x-2-(x-3)}dx`
`+int_(3)^(4) (x-1+x-2+x-3)dx`
`=int_(1)^(2) (-x+4)dx+int_(2)^(3) x dx +int_(3)^(4) (3x-6)dx`
`=[(-x^(2))/(2)+4x]_(1)^(2)+[(x^(2))/(2)]_(2)^(3) +[(3x^(2))/(2)-6x]_(3)^(4)`
`=((-2^(2))/(2)+8)-((-1)/(2)+4)+(1)/(2)(3^(2)-2^(2))`
`+((3)/(2)+4^(2)-6xx4 )-((3)/(2)xx3^(2)-6xx3)`
`=6-(7)/(2)+(5)/(2)+(24-24)-((9)/(2))`
`=(12-7+5)/(2)+0+(9)/(2)=(19)/(2)`
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