Question

Integration of x^/(x-1)(x-2)(x-3)

Answer 1

First we make partial fractions and then integrate the LHS.

x² / [(x-1) (x-2) (x-3)]
  = A/(x-1) + B/(x-2) + C / (x-3)
  = [A (x² - 5x + 6) + B (x² - 4x +3 ) + C(x² - 3x + 2) ] / [(x-1)(x-2)(x-3) ]

x² = A (x² - 5x + 6) + B (x² - 4x +3 ) + C(x² - 3x + 2) 

Let x = 1,    1 = 2 A,  A = 1/2
Let x = 2,    4 = - B,    B = -4
Let x = 3,    9 = 2 C ,   C = 9/2

So LHS = 1/2 * 1/(x-1) -4 / (x-2) + 9/2 * 1/(x-3)

Integrating them now wrt x,

Answer:   1/2 Ln | x-1 | - 4 Ln |x-2| + 9/2 Ln |x-2| + K

We can also combine the various terms.

Answer 2

this is correct answer