Evaluate: `int_0^1|5x-3|dx` (ii) `int_0^pi|cosx|dx` (iii) `int_(-5)^5|x-2|dx` (iv) `int_(-1)^1e^(|x|)dx` (v) `int_0^2|x^2+2x-3|dx` (v) `int_1^4(|x-1|+|x-2|+|x-3|)dx` (vi) `int_(-1)^2|x^3-x|dx`

Explanation

`|5x -3 |={underset(-(5x -3),xlt 3//5)(5x -3, x ge 3//5)`
`:. int_(0)^(1) |5x -3|dx =int_(0)^(3//5) |5x -3|dx`
` +int_(3//5)^(1)|5x -3|dx`
`=int_(0)^(3//5) -(5x -3) dx+ int_(3//5)^(1)(5x -3) dx`
` =-[(5x^(2))/(2) -3x ]_(0)^(3//5) +[(5x^(2))/(2)-3x]_(3//5)^(1)`
`=-[(5)/(2) xx(9)/(25) -3 xx(3)/(5) -0]`
` + [((5)/(2)-3 )-((5)/(2)xx(9)/(25) -3xx(3)/(5))]`
`=-(9)/(10)+(9)/(5)-(1)/(2)-(9)/(10)+(9)/(5) =-(9)/(5) =-(18)/(10)+(18)/(5) -(1)/(2)`
` -(-18+36-5)/(10) -(13)/(10).`