Question

How do you find the antiderivative of `f(x)=(2x^2-5x-3)/(x-3)`?

Answer 1

`x^2+x+C`

Explanation:

`x-3` is a common factor: `f(x)=((2x+1)(x-3))/(x-3)`
`=2x+1`. Hence the antiderivative is `x^2+x+C`.
In general with rational fractionals like this there is no common factor and you have first to do the algebraic division (or equivalent) to get the power of the highest term in the numerator to be less than the highest power in the denominator, plus terms which are constant, `x`, `x^2` etc.