Integration of rational functions

[Kuja]

How do I solve the integration of a rational function such as:

x^2 - 6x - 2
(x^2 + 2)^2

If possible, please list the general rule of solving, I DO NOT want the answer, I simply want to know the way of solving it.
Thanks in advance!

 

[HallsofIvy]

The "rule" that you want is "partial fractions".

If you have a rational function, in which the denominator can be factored into distinct linear factors, such as
(x- 3)/((x+1)(x-2)), then you can write it as a sum of fractions, each having one factor as denominator:
(x-3)/((x+1)(x-2))= A/(x+1)+ B/(x-2).
(Of course, the numerator is of lower degree than the denominator: if not, divide first.)

If some of the linear factors are repeated, such as
(x+ 4)/((x+1)²(x-2)), then you will need all powers of that repeated factor: A/(x+1)+ B/(x+1)²+ C/(x-2)

If some of the factors are quadratics that cannot be factored, then they can, by completing the square, be written in the form "a(x-b)²+ c" and you will need a fraction of the form (Ax+ B)/(a(x-b)²+c), for example (3x²- 2x+ 4)/((x²+ 4)(x+3)) can be written (Ax+B)/(x²+4)+ C/(x+3).

In this particular example,
$$\frac{x^2-6x- 2}{(x^2+2)^2)^2}$$
can be written in the form
$$\frac{Ax+B}{(x^2+2)^2}+\frac{Cx+D}{x^2+2}$$

Those have to be equal for all x so one way of finding A, B, C, D is by setting those equal:
$$\frac{x^2-6x- 2}{(x^2+2)^2)^2}= \frac{Ax+B}{(x^2+2)^2}+\frac{Cx+D}{x^2+2}$$
Now multiply both sides by that denominator to clear the fractions and set x equal to 4 different numbers to get 4 equations for A, B, C, and D. You can often choose those numbers to simplify the equations.

 

[M98Ranger]

To solve the integration of a rational function, such as the one given in the question, you can follow these steps:

1. Factor both the numerator and denominator of the rational function. In this case, the numerator can be factored as (x-2)(x+1) and the denominator can be factored as (x^2+2)^2.

2. Rewrite the rational function as a sum of partial fractions. In this case, the rational function can be rewritten as A/(x-2) + B/(x+1) + C/(x^2+2) + D/(x^2+2)^2, where A, B, C, and D are constants to be determined.

3. Determine the values of A, B, C, and D by equating the coefficients of the rational function with the partial fractions. In this case, you will end up with a system of equations that can be solved to find the values of the constants.

4. Once you have determined the values of A, B, C, and D, you can rewrite the rational function as A/(x-2) + B/(x+1) + C/(x^2+2) + D/(x^2+2)^2.

5. Now, you can integrate each partial fraction separately using the basic integration rules. For example, the integral of A/(x-2) can be found by using the substitution method, while the integral of C/(x^2+2) can be found by using the inverse tangent substitution.

6. Finally, you can combine the integrals of each partial fraction to find the overall integral of the rational function.

The general rule for solving the integration of a rational function is to factor the numerator and denominator, rewrite the rational function as a sum of partial fractions, determine the values of the constants, and then integrate each partial fraction separately before combining them to find the overall integral.