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

Feb 8, 2016

$\frac{5}{2} \ln | x - 2 | - \frac{3}{2} \ln | x | + c$

#### Explanation:

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

hence $\frac{x + 3}{\left(x - 2\right) x} = \frac{A}{x - 2} + \frac{B}{x}$

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 $\Rightarrow B = - \frac{3}{2}$

let x = 2 in (1) : 5 = 2A $\Rightarrow A = \frac{5}{2}$

$\Rightarrow \frac{x + 3}{\left(x - 2\right) x} = \frac{\frac{5}{2}}{x - 2} - \frac{\frac{3}{2}}{x}$

$\Rightarrow \int \frac{x + 3}{\left(x - 2\right) x} \mathrm{dx} = \frac{5}{2} \int \frac{\mathrm{dx}}{x - 2} - \frac{3}{2} \int \frac{\mathrm{dx}}{x}$

$= \frac{5}{2} \ln | x - 2 | - \frac{3}{2} \ln | x | + c$

where c , is the constant of integration.