Power rule, Product rule, Quotient rule, Chain rule
Lists of integrals
Improper integrals
Integration by:
parts, disks, cylindrical
shells, substitution,
trigonometric substitution,
partial fractions, changing order
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 1stdegree polynomial or a power of such a polynomial, then the numerator is a constant. If the denominator is a 2nddegree polynomial or a power of such a polynomial, then the numerator is a 1stdegree polynomial.
Contents
A 1stdegree polynomial in the denominator
The substitution u = ax + b, du = a dx reduces the integral
to
A repeated 1stdegree polynomial in the denominator
The same substitution reduces such integrals as
to
An irreducible 2nddegree polynomial in the denominator
Next we consider such integrals as
The quickest way to see that the denominator x^{2} − 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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