Antidifferentiation of Polynomial Functions Revision Notes for VCE SSCE Mathematical Methods

Antidifferentiation of Polynomial Functions

What is antidifferentiation?

Antidifferentiation is the reverse process of differentiation. When we differentiate $x^2$ , we get $2x$ . Working backwards, if we know that a function has a derivative of $2x$ , we can determine that the original function was $x^2$ . This reverse process is called antidifferentiation.

infoNote

However, there's an important catch: antidifferentiation doesn't give us a single unique answer. Consider these two functions:

f(x) = x^2 + 1

g(x) = x^2 - 7

When we differentiate both functions, we get:

f'(x) = 2x

g'(x) = 2x

Both functions have the same derivative!

chatImportant

If two functions have the same derivative, they differ only by a constant.

The family of antiderivatives

Since many functions can have the same derivative, we call any function whose derivative is $2x$ an antiderivative of $2x$ . The diagram below shows several different antiderivatives of $2x$ :

Each parabola is a vertical translation of $y = x^2$ . They all belong to the same family of curves, and each one differs from the others by a constant amount.

Notation for antiderivatives

We write the general antiderivative of $2x$ as:

\int 2x \, dx = x^2 + c

This notation uses the integral symbol (∫) introduced by Leibniz. We read this as "the indefinite integral of $2x$ with respect to $x$ equals $x^2$ plus $c$ " or "the general antiderivative of $2x$ with respect to $x$ is $x^2 + c$ ".

infoNote

The constant $c$ is called the constant of integration and can be any real number. This single expression represents the entire family of antiderivatives.

General notation

More generally, if we know that $F'(x) = f(x)$ , then:

\int f(x) \, dx = F(x) + c

where $c$ is an arbitrary real number (the constant of integration).

Rules for antidifferentiation

The power rule

From differentiation, we know that if $f(x) = x^n$ , then $f'(x) = nx^{n-1}$ . Reversing this process gives us the power rule for antidifferentiation:

chatImportant

Power Rule for Antidifferentiation:

\int x^n \, dx = \frac{x^{n+1}}{n+1} + c, \quad n \in \mathbb{N} \cup \{0\}

In other words: increase the power by one, then divide by the new power.

For example:

\int x^2 \, dx = \frac{x^3}{3} + c

\int x^7 \, dx = \frac{x^8}{8} + c

\int 1 \, dx = x + c

(since $1 = x^0$ , we get $\frac{x^1}{1} = x$ )

Linearity properties

Antidifferentiation follows three important linearity rules that mirror the rules for differentiation:

Sum rule:

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

Difference rule:

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

Constant multiple rule:

\int k \cdot f(x) \, dx = k \int f(x) \, dx

where $k$ is any real number.

infoNote

These rules mean we can antidifferentiate term by term in polynomial expressions.

Worked examples

lightbulbExample

Worked Example: Finding the General Antiderivative of a Power Function

Find the general antiderivative of $3x^5$ .

Solution:

\int 3x^5 \, dx = 3 \int x^5 \, dx

(taking the constant outside)

= 3 \times \frac{x^6}{6} + c

(applying the power rule)

= \frac{x^6}{2} + c

lightbulbExample

Worked Example: Antidifferentiating a Polynomial

Find the general antiderivative of $3x^2 + 4x^3 + 3$ .

Solution:

\int (3x^2 + 4x^3 + 3) \, dx = 3\int x^2 \, dx + 4\int x^3 \, dx + 3\int 1 \, dx

= \frac{3x^3}{3} + \frac{4x^4}{4} + \frac{3x}{1} + c

= x^3 + x^4 + 3x + c

Notice that we only need one constant $c$ at the end, not a separate constant for each term.

Finding specific antiderivatives

When we're given additional information about a function, we can find a specific antiderivative instead of the general family. This involves using the extra information to determine the value of the constant $c$ .

lightbulbExample

Worked Example: Using a Boundary Condition to Find a Specific Antiderivative

It is known that $f'(x) = x^3 + 4x^2$ and $f(0) = 0$ . Find $f(x)$ .

Solution:

First, find the general antiderivative:

\int (x^3 + 4x^2) \, dx = \frac{x^4}{4} + \frac{4x^3}{3} + c

So $f(x) = \frac{x^4}{4} + \frac{4x^3}{3} + c$ for some constant $c$ .

Now use the condition $f(0) = 0$ :

f(0) = \frac{0^4}{4} + \frac{4(0)^3}{3} + c = 0

c = 0

Therefore:

f(x) = \frac{x^4}{4} + \frac{4x^3}{3}

lightbulbExample

Worked Example: Finding a Curve from its Gradient

If the gradient of the tangent at a point $(x, y)$ on a curve is given by $5x$ , and the curve passes through $(0, 6)$ , find the equation of the curve.

Solution:

Let the curve have equation $y = f(x)$ . We're told that $f'(x) = 5x$ .

Find the general antiderivative:

\int 5x \, dx = \frac{5x^2}{2} + c

So $f(x) = \frac{5x^2}{2} + c$

This describes a family of curves. We need to find which specific member passes through $(0, 6)$ .

Since the curve passes through $(0, 6)$ , we have $f(0) = 6$ :

f(0) = \frac{5(0)^2}{2} + c = 6

c = 6

Therefore:

f(x) = \frac{5x^2}{2} + 6

lightbulbExample

Worked Example: Finding y in Terms of x

Find $y$ in terms of $x$ if $\frac{dy}{dx} = x^2 + 2x$ , and $y = 1$ when $x = 1$ .

Solution:

Find the general antiderivative:

\int (x^2 + 2x) \, dx = \frac{x^3}{3} + x^2 + c

So $y = \frac{x^3}{3} + x^2 + c$

Use the condition that $y = 1$ when $x = 1$ :

1 = \frac{1^3}{3} + 1^2 + c

1 = \frac{1}{3} + 1 + c

c = -\frac{1}{3}

Therefore:

y = \frac{x^3}{3} + x^2 - \frac{1}{3}

Using technology

Modern calculators and computer algebra systems can help with antidifferentiation:

infoNote

When using technology:

Select "indefinite integral" to find the general antiderivative
Remember to add the constant $c$ to the answer (calculators often omit this)
To find specific antiderivatives, define the function family and then solve for $c$ using the given condition

Summary

bookmarkSummary

Key Points to Remember:

Antidifferentiation is the reverse of differentiation – it finds a function from its derivative
The constant of integration is crucial – always add $+ c$ when finding indefinite integrals
Functions with the same derivative differ by a constant – this is why we get a family of antiderivatives
The power rule for antidifferentiation: $\int x^n \, dx = \frac{x^{n+1}}{n+1} + c$ (add one to the power, divide by the new power)
Use boundary conditions to find specific antiderivatives – extra information about a point on the curve allows you to determine the value of $c$

Antidifferentiation of Polynomial Functions (VCE SSCE Mathematical Methods): Revision Notes