Partial fractions in integration

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In integral calculus, partial fraction expansions provide an approach to integrating a general rational function. Any rational function of a real variable can be written as the sum of a polynomial function and a finite number of fractions. Each fraction in the expansion has as its denominator a polynomial function of degree 1 or 2, or some positive integer power of such a polynomial. (In the case of rational function of a complex variable, all denominators will have a polynomial of degree 1, or some positive integer power of such a polynomial.) If the denominator is a 1st-degree polynomial or a power of such a polynomial, then the numerator is a constant. If the denominator is a 2nd-degree polynomial or a power of such a polynomial, then the numerator is a 1st-degree polynomial.


A 1st-degree polynomial in the denominator

The substitution u = ax + b, du = a dx reduces the integral


A repeated 1st-degree polynomial in the denominator

The same substitution reduces such integrals as


An irreducible 2nd-degree polynomial in the denominator

Next we consider such integrals as

The quickest way to see that the denominator x2 − 8x + 25 is irreducible is to observe that its discriminant is negative. Alternatively, we can complete the square:

and observe that this sum of two squares can never be 0 while x is a real number.

In order to make use of the substitution

we would need to find x − 4 in the numerator. So we decompose the numerator x + 6 as (x − 4) + 10, and we write the integral as

The substitution handles the first summand, thus:

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