Depending on the situation, you may use this way to write a fraction, whose denominator is a polynomial.
If you need to solve an indefinite integral, splitting the fraction may help you to solve more but easier indefinite integrals.
In this case, the original numerator may be evaluated such that:
It is a way to show that a fraction could be written as a sum of 2 or more fractions, depending on the number of terms from numerator.
In this case, the problem combined 2 terms from the total of 4 terms. In this way, we can add 2 fractions that have the same denominator.
Also the fraction could be written as an algebraic sum of 4 terms.
x/(x^2+1) + 1/(x^2+1) + 2x/(x^2+1) - 2/(x^2+1)
Since the denominators are common, we'll write:
(x + 1 + 2x - 2)/(x^2+1)
We'll combine like terms and we'll get the result:
(3x - 1)/(x^2+1)