Question

How do you integrate `(x+2) / (x(x-4))` using partial fractions?

Answer 1

`\color{red}{\int \frac{x+2}{x(x-4)}\ dx}=\color{blue}{-1/2\ln |x|+3/2\ln|x-4|+C`

Explanation:

Let

`\frac{x+2}{x(x-4)}=A/x+B/{x-4}`

`(A+B)x-4A=x+2`

Comparing the corresponding coefficients on both the sides we get

`A+B=1\ \ &\ \ -4A=2`

Solving above equations we get

`A=-1/2, B=3/2`

Now, the partial fractions can be written as follows

`\frac{x+2}{x(x-4)}=(-1/2)/x+(3/2)/{x-4}`

`=-1/{2x}+3/{2(x-4)}`

`\therefore \int \frac{x+2}{x(x-4)}\ dx`

`=\int (-1/{2x}+3/{2(x-4)})\ dx`

`=-1/2\int dx/x +3/2\int dx/{x-4}`

`=-1/2\ln |x|+3/2\ln|x-4|+C`