How do you integrate #int (x+3) /( (x-2)x)# using partial fractions?

1 Answer
Feb 8, 2016

# 5/2ln|x - 2 | -3/2ln|x| + c#

Explanation:

since the factors in the denominator are linear then the numerators will be constants.

hence # (x+3)/((x-2)x) = A/(x-2) + B/x #

now multiply through by (x-2)x

to obtain : x + 3 = Ax + B(x-2 )...........................(1)

now require to find values of A and B. Note that if x = 0 then the term with A will be zero and if x = 2 the term with B will be zero.

let x = 0 in (1) : 3 = -2B # rArr B = -3/2 #

let x = 2 in (1) : 5 = 2A # rArr A = 5/2 #

# rArr (x+3)/((x-2)x) = (5/2)/(x-2) -( 3/2)/x#

#rArr int(x+3)/((x-2)x) dx = 5/2 intdx/(x-2) - 3/2intdx/x #

# = 5/2ln|x-2| -3/2ln|x| + c#

where c , is the constant of integration.