Antidifferentiation as the reverse of differentiation, the antiderivative of polynomial functions via the power rule, the constant of integration, and the use of an initial condition to determine a specific antiderivative

What this dot point is asking

VCAA wants you to recognise antidifferentiation as the reverse of differentiation, apply the power rule to find antiderivatives of polynomial functions, include the constant of integration, and use an initial condition to determine the specific antiderivative. The dot point is the introduction to integration that Unit 4 will extend.

The reverse of differentiation

If $F'(x) = f(x)$, then $F(x)$ is an antiderivative of $f(x)$.

The general antiderivative is written:

where $C$ is the constant of integration. Because differentiation kills constants, two antiderivatives of the same function can differ by any constant, so $C$ must always be included unless an initial condition fixes it.

The reverse power rule

For polynomial functions, the reverse of $\frac{d}{dx}(x^n) = n x^{n-1}$ is:

The reverse procedure: add 1 to the exponent, divide by the new exponent.

The $n = -1$ case ($\int x^{-1} \, dx$) is excluded because dividing by zero is undefined; the antiderivative of $1/x$ is $\ln|x|$, which is introduced in Unit 4.

Linearity

Antidifferentiation distributes over sums and pulls out constants:

So you can antidifferentiate term by term.

The constant of integration

Every antiderivative requires $+ C$ unless an initial condition fixes it.

Initial value problem. Given $f'(x) = \ldots$ and a specific value $f(a) = b$:

1. Antidifferentiate to get $f(x) = F(x) + C$.
2. Substitute $x = a$: $F(a) + C = b$.
3. Solve for $C$.
4. Write the specific $f(x)$.

The constant is then determined; the general antiderivative has become a specific function.

Worked examples

Example 1. Simple polynomial

$f'(x) = 3 x^2 - 2 x$. Antidifferentiate:

$\int 3 x^2 \, dx = x^3$.

$\int -2 x \, dx = -x^2$.

$f(x) = x^3 - x^2 + C$.

Example 2. With initial condition

$f'(x) = 4 x^3 + 6 x$ and $f(0) = 5$.

$f(x) = x^4 + 3 x^2 + C$.

Apply $f(0) = 5$: $0 + 0 + C = 5$, so $C = 5$.

$f(x) = x^4 + 3 x^2 + 5$.

Example 3. Kinematics application

A particle has velocity $v(t) = 3 t^2 + 2$ m/s, and at $t = 0$ is at the origin. Find the position $x(t)$.

$x(t) = \int v(t) \, dt = t^3 + 2 t + C$.

Apply $x(0) = 0$: $0 + 0 + C = 0$, so $C = 0$.

$x(t) = t^3 + 2 t$ m.

At $t = 3$ s, $x = 27 + 6 = 33$ m.

Connection to Unit 4

Unit 4 will extend antidifferentiation to:

• The antiderivatives of $e^{kx}$, $\frac{1}{x}$, $\sin(kx)$, $\cos(kx)$.
• The definite integral and the Fundamental Theorem of Calculus.
• Area between curves.
• Integration by substitution.

The Unit 2 foundation is the power rule and the discipline of including the constant of integration.

Common errors

Forgetting $+ C$. An indefinite integral without the constant of integration loses marks.

Power rule applied to $1/x$. The power rule for $\int x^n \, dx$ requires $n \neq -1$. The antiderivative of $1/x$ is $\ln|x|$ (Unit 4).

Reversing the rule incorrectly. Differentiate: multiply by $n$, decrease exponent. Antidifferentiate: increase exponent, divide by new exponent. The factor of $n$ goes opposite directions.

Substituting before antidifferentiating. When applying an initial condition, antidifferentiate the function first, then substitute.

Antiderivative as plural. A function has infinitely many antiderivatives differing by a constant. The general antiderivative captures all of them via $+ C$.

In one sentence

Antidifferentiation in Unit 2 is the reverse of differentiation, with the power rule $\int x^n \, dx = x^{n+1}/(n+1) + C$ for polynomial functions (excluding $n = -1$), linearity allowing term-by-term integration, and the constant of integration $C$ always required unless an initial condition determines it; the dot point is the foundation for Unit 4's definite integration and applications.