antiderivative of rational function
Theorem. The antiderivative of a rational function is always expressible in a closed form, which only can comprise, except a rational expression summand, summands of logarithms and arcustangents of rational functions.
One can justify the theorem by using the general form of the (unique) partial fraction decomposition
of the rational function ; here, is a polynomial, the first sum expression is determined by the real zeroes of the denominator of , the second sum is determined by the real quadratic prime factors of the denominator (which have no real zeroes).
The addends of the form in the first sum are integrated directly, giving
and make the substitution
i.e. , getting
where and are certain constants. In the case we have
and in the case
The latter addend of the right hand side of (4) is for got from
and for the cases on may first write
The assertion of the theorem follows from (1), …, (8).
|Title||antiderivative of rational function|
|Date of creation||2013-03-22 19:21:38|
|Last modified on||2013-03-22 19:21:38|
|Last modified by||pahio (2872)|
|Synonym||integration of rational functions|