Here are some of the integrals we've discovered already:
 

The next examples illustrate the key ideas for more general problems.
Example VII.F.1: Find `int (x^2 + 3x + 5)/(x-1) dx`.
Solution: The first thing to notice here is that the polynomial in the numerator of this rational function has degree 2 while the polynomial in the denominator has degree 1. A quotient of polynomials where the degree of the numerator is greater than or equal to the degree of the denominator is called an improper fraction and is simplified by the long division of these two polynomials as follows:

Thus `(x^2 + 3x + 5)/(x - 1) = x + 4 + (9)/(x-1)` and the final integration result can be obtained by using linearity and a simple substitution: [You should fill in the details.]

Thus `(x^3)/(x^2 + 1) = x - (x)/(x^2 + 1)` and we have the result (details omitted )

Now since `v^2 + 4 = 4 ((v/2)^2 +1)` we let `u = v/2`, so `2du = dv`, and we have

This type of identity is called a partial fraction decomposition.

To answer the second question first, it can be shown that any rational function has a partial fraction decomposition that will allow one to integrate the function with relative ease. The proof of this result is found in more advanced work in algebra and will not be given here. Perhaps more important to our purposes is the answer to the first question of how to find such decompositions. This also calls for an excursion into algebra, but one worth the effort at this time. 

Example VII.F.6: Find a partial fraction decomposition for `(3x+1)/(x^2-1)`.
Solution: The method we use might be described as "wishful thinking" analysis. Imagine that we already have a partial faction decomposition. What will it look like? First factor the denominator, `x ^2-1 = (x+1)(x-1)`. Now proceed to try to find constants A and B so that `(3x+1)/(x^2 - 1) = A/(x+1) + B/(x-1)`.

Multiplying both sides of this equation by the denominator `x^2 - 1`, we obtain the equation

We'll give two methods for finding the constants at this stage.

Method 1. Since these polynomials are equal, more wishful thinking suggests the coefficient of `x` and the constant term must also be equal. Thus we have two equations: `A + B = 3` from  the coefficient of `x` and `B - A = 1` from the constants.
Therefore adding these equations gives `2B = 4`, `B = 2`, and then we see that `A = 1`.

Method 2. Since these polynomials are equal, they give the same value for any choice of `x`.
Let `x = 1`. Then `3(1)+1 = A(1 - 1) + B(1 + 1) = 2B`.
So again we find that `4 = 2B` and `B = 2`.
Now let `x = 0`, use `B = 2`and we see that `3(0)+1 = A(0-1) + 2(0 + 1)`, so `1 = -A + 2` and `A = 1`.
 

Comment: Another choice that would be easier than `x = 0` is `x = -1`. Then we have

Multiplying both sides of the equation by the denominator `x^3 - 1`, we obtain the equation

Method 2. Let `x = 1`. Thus `1 = A(1 + 1 + 1) + (B + C)(1 - 1)` and `A = 1/3`.
Let `x = 0`. Then `0 = 1/3(0 + 0 + 1) + (B <sup>.</sup> 0 + C)(0 - 1)`, and `C = 1/3`.
Finally let `x = -1`. Then `-1 = 1/3(1) + (B(-1) + 1/3)(-2)`, or `-2B = -2/3` so that `B = -1/3`.

Note: In this example we can combine these methods. From Method 1 we see that `A = C = -B`. Using `x=1` in Method 2 we find that  `A= 1/3` , hence `A = C = 1/3`, and `B = -1/3`. 

First, you need to factor the denominator of the proper rational function into a product of linear and quadratic terms. It is a fundamental theorem of algebra (usually proven in an advanced course in mathematics) that this is possible in theory. Note that even though the theory says such a factoring is possible, there is another side to the theory that says that there is no direct method for factoring a polynomial of degree five or greater. So there is a sizeable gap between the theory of factoring and its practice.

After factoring the denominator, you then write an expression involving numerous arbitrary constants as coefficients of simpler terms with denominators determined by the factoring. The actual terms are described below in Tables VII.F.i and VII.F.ii. Once this expression is written, you equate it with the original function and multiply by the denominator to give an equation of polynomials. You can now use one of the two methods illustrated above to solve for the constants in this equation.

COMMENT: This technique will give a partial fraction decomposition for any proper rational function. The integration of a rational function can be reduced using this decomposition to the integration of each of the terms of the decomposition.

(See Seeley, Calculus of One Variable, pgs.234-5)

STEP 1. Long Division. If you want to integrate `(P(x))/(Q(x))` where `P(x)` and `Q(x)` are polynomials,
write `(P(x))/(Q(x)) = T(x) + (R(x))/(Q(x))` where `T(x)` and `R(x)` are polynomials and the degree of `R(x)` is less than the degree of `Q(x)`. DO THIS BY USING LONG DIVISION OF POLYNOMIALS. This reduces the problem to integrating `(R(x))/(Q(x))` because `int T(x) dx` is easy. [See Ex.VII.F.1.]

STEP 2. Factor the denominator. This will be a product of linear terms and quadratic terms. Complete the squares of the quadratic (nonfactorable) terms. Thus you'll have

[The factoring of polynomials in this fashion is a very important result of algebra usually referred to as one form of the "Fundamental Theorem of Algebra." It was first proven by Gauss in his doctoral dissertation, and is proven in several advanced courses in mathematics. ]

STEP 3.  Write the answer. Write the integral `int (R(x))/(Q(x)) dx` as a sum of terms, each term multiplied by an undetermined constant and arising from one of the factors of Q(x) according to the following rules:

(a) Each factor of the form `(x - a)^r` requires the terms:

(b) Each factor of the form `[(x-b)^2+c^2]^s` requires the terms:

Remark: Step 3 is the most important since now `int (R(x))/(Q(x)) dx` is a linear combination of functions of the types listed in (a) and (b).

What you must remember is the types of terms required by the factors of `Q(x)`.
The number of terms in this combination is equal to the degree of `Q(x)`.

STEP 4. Differentiate.  Find the derivative of the the linear combination written in step 3 and place the answer over the common denominator, which is `Q(x)`! [Don't forget the chain rule!] The numerator is a polynomial of degree one less than the degree of `Q(x)`.

Don't spend any time with multiplying out the denominator!

STEP 5. Solve. Equate the numerator found in step 4 with `R(x)`. Now determine the constants introduced in step 3 by using one or both of the following methods.

(a) Multiply out the numerator with the undetermined coefficients, write it in standard polynomial form, and equate the coefficients of the corresponding powers of `x`. This will give a system of linear equations which can be solved by any of several techniques of linear algebra.

(b) Substitute some particular values for `x` in the numerator and `Q(x)` to obtain simpler expressions relating the undetermined coefficients.[See examples VII.F.... above.]

Solution: We follow Seeley's method.
Step 1. Long Division. We notice that the integrand is a proper rational function so we may proceed to
Step 2. Factor the denominator. Completing the square of the quadratic factor we have

so the denominator is `[(x - 3)^2 + 2^2] (x - 2)^2` .

Step 3. Write the answer. We can now say the integral is the following combination of functions determined by the factors of the denominator:

Note that the denominator is a fourth degree polynomial and the previous combination involves exactly four summands with four constants to be determined.

Step 4. Differentiate. The derivative of the last expression is equal to the integrand [`(x + 1)/((x^2 - 6x + 13)(x-2)^2)`].

Well this is certainly a complicated expression, but it does have the same common denominator as the integrand. After some further algebra we arrive at
Step 5. Solve. We equate  the numerator of the integrand and the numerator of the last expression when placed over the common denominator and solve for the constants.

This is definitely one long mess!
Letting `x = 2` gives some help, so that `3 = -B ( 4 -12 + 13)` and thus `B = -3/5`.
Letting `x = 3`, 1,and 0 gives us three equations that assist in completing the algebraic part of the problem.

Let `x = 3` :    `4 = A(1)(4) + 3/5 (4) + 2C(1)` ; so `8/5 = 4 A + 2 C`.

Let `x = 1` :    `2 = A(-1)(8) + 3/5 (8) + 2C + 2(-2)D(1)`; so ` -14/5 = -8A + 2 C - 4 D`.

Let `x = 0` :    `1 = A(-2)(13) + 3/5 (13) + 2C(4) + (2)(-3)D(4)`; so `-34/5 = -26 A + 8 C -24 D`.

One can check that the solution to these equations is given by `A = 11/25, C = -2/25`  and `D = -11/50`.
Thus

Find the following integrals:

Here are some more: