Polynomial Function

The first step in knowing how to integrate a rational function is understanding what a polynomial function is. A polynomial function is any function made up of one or more monomials.
 

Type Definition Example
A Monomial One monomial Constant
B Binomial Two monomials Linear function
C Trinomial Three monomials Quadratic function
polynomial_examples
All of the examples given above are examples of polynomials. A polynomial function can have any combination of monomials.

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Rational Function

 

A rational function is defined as a function that divides one polynomial by another. This is easy to remember because of the word rational.

ratio_example

Above, you can see that a rational function is simply the ratio of two polynomial functions. Keep in mind that even regular polynomials can be rational functions.

ratio_explanation

By dividing the polynomial by 1, we get a ratio of any polynomial function and a monomial. A rational function can be simple, like in the examples above, or more complex. The table below has some examples of more complex rational functions.

 

Example Numerator Denominator
\frac{2x+4}{3x^2+2x+5} Linear Quadratic
\frac{\sqrt{3x+2}}{4x} Square root of linear Linear
\frac{4x^2+10x+2}{5x^2+8x+9} Quadratic Quadratic

 

Rational Function Simplification

Whenever you’re dealing with rational functions, the first thing you have to do is simplify them. Let’s look at an easy example first.

fraction_example

Whether we’re graphing this rational function or integrating it, we should always try to simplify it. This will make it easier to perform any operation on it afterwards. The simplification steps for this rational function are in the table below.

 

Description Problem
Step 1 See if the numerator and denominator have common terms There is an x term in both
Step 2 Split up the numerator \frac{3x}{x} + \frac{4}{x}
Step 3 Cancel out any terms that are alike \frac{4}{x} + 3

 

Here are some more tips you can use to simplify any rational function.

 

Example
See if the numerator or denominator have any like terms \frac{3x+5}{10x}
Test whether the numerator or denominator can be simplified by themselves \frac{x^2 + 4x + 4}{x+2} = \frac{(x+2)(x+2)}{x+2}
Add like terms \frac{3x + 4x + 5}{2 + 2x +4} = \frac{7x +5}{2x + 6}

 

Integration

Integration is one of the most important concepts in calculus. You can think about integration in two ways:

 

1 Integration is the opposite of taking the derivative \int f(x) dx
2 Integrating a function gives us the equation for the area \int f(x) dx

 

Notice that the notation is the same for both points of view. Whether you think about it in the first or second way, the process of integration is always the same. You may also see the notation like this:

integration_elements
A Integral sign The sign for integration
B Function The function we want to integrate
C Lower bound The lower limit of the interval we want to find the area for
D Upper bound The upper limit of the interval we want to find the area for
E dx Specifies the variable to integrate

 

Integration of a Fraction

There are several ways you can integrate a fraction. First, take a look at some basic integration rules.

 

Integration Function Result
Constant \int a dx ax
Power \int x^{n} dx \frac{x^{n+1}}{n+1}
Reciprocal \int \frac{1}{x} dx ln|x|

 

Take a look at some examples below.

 

Integration Result
Constant \int 3 dx 3x
Power \int x^{4} dx \frac{x^{4+1}}{4+1} = \frac{x^{5}}{5}
Reciprocal \int \frac{1}{2x} dx ln|2x|

 

This means that, depending on what fraction you have, you can integrate several ways. The first method is to use the following rule.

power_inverse

Let’s take the following fraction as an example.

fraction_exponent

To make this function easier to integrate, we can use the rules of powers to get the following:

power_negative

The second method is to use the reciprocal rule. In many instances, we can combine this with u-substitution:

u_substitution_integration

 

Integration of Rational Function

Integrating rational function requires using all the techniques mentioned above. You may encounter more complex functions, which have the following rules:

 

Integration Function Result
Log \int ln(x) dx x * ln(x) - x
Cosine \int cos(x) dx sin(x)
Sine \int sin(x) dx -cos(x)
Tangent \int tan(x) dx tan(x)

 

Take a look at some examples below.

 

Example Result
Log \int ln(3x) dx 3x * ln(3x) - 3x
Cosine \int cos(45) dx sin(45)
Sine \int sin(20) dx -cos(20)

 

Since rational functions are fractions, we can use power rules like in the examples before. However, you will also need u-substitution and integration by parts.

multiplication_rule_integration

Where:

 

A Function f(x)
B Function g(x)
C Derivative of f(x)

 

Let’s take an example.

integration_cos

Using integration by parts, we do the following:

u_substitution_integration_example

Now, we simply follow the integration rules.

integration_parts_rules

Now, we simplify.

sin_cos_integration_parts

 

Example 1

Let’s do a step-by-step integration of a rational function. Take the function below as an example.

 

    \[ \int x^{3} dx \]

 

Here, we can simply use the power integration rule. First, use the rule to find the result.

 

    \[ \int x^{3} dx = \frac{x^{3+1}}{3+1} \]

 

Next, simplify the equation to get the final result.

 

    \[ \frac{x^{3+1}}{3+1} = \frac{x^{4}}{4} \]

 

Example 2

In the last example, you worked with an indefinite integral. Let’s take the same example, but work with a definite integral instead.

 

    \[ \int  \limits_3^5 x^{3} dx \]

 

We work with the results from the previous example.

 

    \[ \int  \limits_3^5 \frac{x^{4}}{4} = \frac{5^{4}}{4} - \frac{3^{4}}{4} = 136 \]

 

Example 3

Let’s work with u-substitution in this example. You have the following integral.

 

    \[ \int \sqrt{3x + 1} dx \]

 

Let’s substitute 3x+1 with u. So, let’s find the derivative of the u term.

 

    \[ du = 3 dx \]

 

Solve for the dx term.

 

    \[ dx = \frac{du}{3} = \frac{1}{3}*du \]

 

Replace these terms into the original integral.

 

    \[ \int \sqrt{u} du \frac{1}{3} \]

 

Simplify this integral.

 

    \[ \frac{1}{3} \int u^{\frac{1}{2}} du \]

 

Solve using the power rule.

 

    \[ \frac{1}{3} (\frac{u^{\frac{3}{2}}}{\frac{3}{2}}) \]

 

Plug u term back in.

    \[ \frac{1}{3} (\frac{(3x+1)^{\frac{3}{2}}}{\frac{3}{2}}) \]

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Danica

Located in Prague and studying to become a Statistician, I enjoy reading, writing, and exploring new places.