Suppose that, #I=int(x^2-9x+14)/{(x^2-4x+3)(x-4)}dx#.
# :. I=int(x^2-9x+14)/{(x-3)(x-1)(x-4)}dx#.
We will decompose the Integrand #(x^2-9x+14)/{(x-3)(x-1)(x-4)}#
using the Method of Partial Fraction.
To this end, we let, for some #A,B,C in RR#,
# (x^2-9x+14)/{(x-3)(x-1)(x-4)}=A/(x-1)+B/(x-3)+C/(x-4)#.
We use Heaviside's Cover Up Method to determine #A,B,C in RR#.
#:. A=[(x^2-9x+14)/{(x-3)(x-4)}]_(x=1)=(1-9+14)/{(-2)(-3)}=1#,
# B=[(x^2-9x+14)/{(x-1)(x-4)}]_(x=3)=(9-27+14)/{(2)(-1)}=2#,
# C=[(x^2-9x+14)/{(x-3)(x-1)}]_(x=4)=(16-36+14)/{(1)(3)}=-2#.
#:. I=int{1/(x-1)+2/(x-3)-2/(x-4)}dx#,
#=ln|(x-1)|+2ln|(x-3)|-2ln|(x-4)|#.
# rArr I=ln|(x-1)|+2ln|(x-3)|-2ln|(x-4)|+C, or, #
# I=ln|{(x-1)(x-3)^2}/(x-4)^2|+C.#
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