Integration of rational fraction:
In order to convert improper rational function into a proper one, we can use long division:
where F(x) is a polynomial, R(x)Q(x) is a proper rational function.
To integrate a proper rational function, we can apply the method of partial fractions.
This method allows to turn the integral of a complicated rational function into the sum of integrals of simpler functions.
The denominators of the partial fractions can contain non-repeated linear factors, repeated linear factors, non-repeated irreducible quadratic factors, and repeated irreducible quadratic factors.
To evaluate integrals of partial fractions with linear or quadratic denominators, we use the following 6 formulas:
For partial fractions with irreducible quadratic denominators, we first complete the square:
Hence, we can write:
The possible cases for fractions with quadratic denominators are covered by the following integrals:
Finally, the integral ∫dt(t2+m2)k can be evaluated in k steps using the reduction formula