Question

`int_2^3 (2x+1)/(x^3 - 5x^2 + 4x)dx`?

Answer 1

`-1.11164`

Explanation:

`"This is the integral of a rational function."`
`"The standard procedure is splitting in partial fractions."`
`"First, we search for the zeros of the denominator :"`
`x^3 - 5 x^2 + 4 x = 0`
`=> x (x - 1)(x - 4) = 0`
`=> x = 0, 1, or 4`

`"So we split in partial fractions : "`
`(2x + 1)/(x^3-5x^2+4x) = A/x + B/(x-1) + C/(x-4)`
`=>2x + 1 = A(x-1)(x-4) + B x(x-4) + C x(x-1)`
`=> A+B+C = 0 , -5 A - 4 B - C = 2, 4A = 1`
`=> A=1/4 , B = -1, C = 3/4`

`"So we have"`

`(1/4) int {dx}/x - int {dx}/(x-1) + (3/4) int {dx}/(x-4)`
`= (1/4) ln(|x|) - ln(|x-1|) + (3/4) ln(|x-4|) + C`

`"Now we evaluate between 2 and 3 :"`

`= (1/4) ln(3) - ln(2) + cancel((3/4) ln(1)) - (1/4) ln(2) + cancel(ln(1)) - (3/4) ln(2)`
`= (1/4) ln(3) - 2 ln(2)`
`= -1.11164`