Integration of polynomials

How do you integrate polynomial functions, where different terms have different powers and coefficients? This skill is important for calculating projectile trajectories in physics, beam deflections in engineering and determining cost functions and profit maximisation in economics. Use this resource to learn how.

In Antidifferentiation, you learned how to find the antiderivative of functions consisting of single algebraic terms. Here, we will look at polynomials which have two or more terms.

Polynomial functions have the general form:
\[ f(x)=a_{0}+a_{1}x+a_{2}x^{2}+\cdots+a_{n}x^{n} \]

where \(a_{i},\,i=1,2,3,\ldots,n\) are constants.

Linearity rules

There are two properties that make it easier for us to find the integrals of more complex functions. They are called the linearity rules.

A constant that multiplies a function can be taken out of the integral. For a constant \(a\):

\[\int af(x)dx = a\int f(x)dx\]

The other rule states that when we differentiate the sum or difference of two functions \(f(x)\) and \(g(x)\), we can differentiate each function separately then add or substract the results.

\[\int (f(x)+g(x))dx = \int f(x)dx + \int g(x)dx\] \[\int (f(x)-g(x))dx = \int f(x)dx - \int g(x)dx\]

You can have any number of terms in the integral. For example, if \(a_{1},a_{2},a_{3}\ldots a_{n}\) are constant:

Example 1 – using the linearity rules

Find \(\int(5x^{2}-3x+4)dx\).

We can use the linearity rules to separate these into two separate integrals.
\[\int(5x^{2}-3x+4)dx = \int5x^{2}dx+\int(-3x)dx\int4dx\]

We can now use the linearity rules to take the constants out of the integrals.
\[= 5\int x^{2}dx - 3\int xdx+4\int dx\]

Now, we are ready to antidifferentiate.
\[= \frac{5}{3}x^{3}-\frac{3}{2}x^{2}+4x+c\]

There is a constant for each of the integrals on the right hand side. However, a constant plus a constant still gives you a constant so we just use a single constant \(c\in\mathbb{R}\).

Exercise – using the linearity rules

Integrate the following with respect to \(x\).

1. \(x^{3}-2x\)
2. \(2x^{4}+5x^{2}\)
3. \(3x^{2}-7x^{5}\)
4. \(x^{2}-2+x\)
5. \(2x+2\)
6. \(-6x^{3}+5x+2\)
7. \(9x^{6}-3x-4\)

Definite integrals

So far, we have looked at functions of \(x\) where the antiderivative involved a constant \(c\). These integrals are called indefinite integrals. They do not have a specific value.

If we define a definite integral for a function \(f(x)\), from \(x=a\) to \(x=b\) with respect to \(x\), we use the following notation:

\[ \int_{a}^{b}f(x)dx=\left[F(x)\right]_{a}^{b}=F(b)-F(a) \]

where we assume \(f(x)\) is defined for all \(x\) in the interval \([a,b]\). \(a\) the lower limit of integration and \(b\) the upper limit of integration. \([a,b]\) is called the limits of integration.

The notation \(\left[F(x)\right]_{a}^{b}\) means to substitute \(x=b\) and \(x=a\) into \(F(x)\) and then subtract the values. This gets rid of the constant \(c\).

To find the definite integral:

1. Find the antiderivative or indefinite integral \(\int xdx+c\).
2. Substitute \(x=b\) into the integral, i.e. evaluate this at the upper limit of integration.
3. Substitute \(x=a\) into the integral, i.e. evaluate this at the lower limit of integration.
4. Minus the integral at the lower limit from the integral at the upper limit. This lets us cancel out the constant \(c\).

This is known as the first fundamental theorem of calculus. Let's look at some examples of how this is applied.

Example 1 – finding definite integrals

Find \(\intop_{1}^{3}xdx\).

First, we find an antiderivative or indefinite integral \(\int xdx+c\).

An antiderivative is:
\[\frac{1}{2}x^{2}+c\]

Now, we can evaluate this at the limits of integration.
\[\begin{align*} \int_{1}^{3}xdx & = \left[\frac{1}{2}x^{2}+c\right]_{x=1}^{x=3},\,c\in\mathbb{R}\\
& = \left(\frac{1}{2}3^{2}+c\right)-\left(\frac{1}{2}1^{2}+c\right)\\
& = \left(\frac{9}{2}+c\right)-\left(\frac{1}{2}+c\right)\\
& = \frac{8}{2}\\
& = 4
\end{align*}\]

Exercise – finding definite integrals

Evaluate the following indefinite integrals:

1. \(\intop_{1}^{2}(6x-2)dx\)
2. \(\intop_{2}^{3}(8x^{3}-9x^{2}-2x)dx\)
3. \(\intop_{0}^{2}(7x^{2}+2)dx\)
4. \(\intop_{-1}^{2}(7x^{3}-6x+3)dx\)
5. \(\intop_{-1}^{1}(8x^{3}-x^{2}+6)dx\)
6. \(\intop_{-1}^{0}(8x^{3}-x^{2}+6)dx\)