Question

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

Answer 1

`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.