Antiderivatives and indefinite integration

IB Mathematics Applications and Interpretation HL· 5 min read

1. Antiderivatives as the inverse of differentiation★★☆☆☆⏱ 15 min

Differentiation calculates the rate of change of a function; integration reverses this process. If $F^{'} (x) = f (x)$ , then $F (x)$ is an antiderivative of $f (x)$ . All antiderivatives of $f (x)$ can be written as $F (x) + C$ , where $C$ is any real constant called the constant of integration.

📘 Definition

Antiderivative

$F (x) + C, C \in R$

A function $F (x)$ is an antiderivative of $f (x)$ on an interval if $F^{'} (x) = f (x)$ for all $x$ in that interval.

📐 Worked Example

Verify that $F (x) = x^{3} + 2 x + 5$ is an antiderivative of $f (x) = 3 x^{2} + 2$ .

1
Differentiate $F (x)$ term-by-term with respect to $x$ :
2
$\frac{d}{dx}\left(x^3 + 2x + 5\right) = 3x^2 + 2 + 0$
3
The derivative of $F (x)$ matches the original function $f (x)$ , so $F (x)$ is an antiderivative. The constant $+ 5$ is just one specific value of the arbitrary constant of integration $C$ .

Exam tip:

Missing the $+ C$ at the end of an indefinite integral answer almost always costs an entire mark, it is one of the most common exam errors.

2. Standard integration rules★★★☆☆⏱ 20 min

We derive integration rules by reversing common differentiation rules. The table below lists all standard rules for indefinite integration required for IB AI HL:

Function $f (x)$	Indefinite integral $\int f (x) d x$
$x^{n} (n \neq = - 1)$	$\frac{x ^{n + 1}}{n + 1} + C$
$\frac{1}{x} = x^{- 1}$	$ln ∣ x ∣ + C$
$e^{k x}$	$\frac{1}{k} e^{k x} + C$
$sin (k x)$	$- \frac{1}{k} cos (k x) + C$
$cos (k x)$	$\frac{1}{k} sin (k x) + C$
$sec^{2} (k x)$	$\frac{1}{k} tan (k x) + C$

📐 Worked Example

Calculate $\int (4 x^{3} - 2 sin 2 x + \frac{3}{x}) d x$ .

1
Integrate each term separately using the standard rules:
2
First term (power rule):
3
$\int 4x^3 dx = 4 \cdot \frac{x^4}{4} = x^4$
4
Second term (trigonometric rule):
5
$\int -2\sin 2x dx = -2 \cdot \left(-\frac{1}{2}\cos 2x\right) = \cos 2x$
6
Third term (reciprocal rule):
7
$\int \frac{3}{x} dx = 3\ln|x|$
8
Add the constant of integration for the final answer:
9
$\int \left(4x^3 - 2\sin 2x + \frac{3}{x}\right) dx = x^4 + \cos 2x + 3\ln|x| + C$

3. Finding the constant of integration★★★☆☆⏱ 20 min

For most applied problems, we are given extra information called an initial condition that lets us find the specific value of $C$ , instead of leaving it as an arbitrary constant. This gives us a unique particular solution that satisfies the given condition.

📘 Definition

Initial Value Problem

A problem that asks for a specific antiderivative, given the derivative of the function and the value of the function at a given point.

📐 Worked Example

A curve has derivative $\frac{d y}{d x} = 3 x^{2} - 2 x$ , and $y = 4$ when $x = 2$ . Find the equation of the curve.

1
First find the general indefinite integral (with $C$ ):
2
$y = \int (3x^2 - 2x) dx = x^3 - x^2 + C$
3
Substitute the initial condition $x = 2, y = 4$ to solve for $C$ :
4
$4 = (2)^3 - (2)^2 + C \implies 4 = 8 - 4 + C \implies C = 0$
5
Substitute $C$ back to get the final particular solution:
6
$y = x^3 - x^2$

✓ Quick check

Test your understanding:

What is the value of $C$ if $\int 2 x d x = x^{2} + C$ and the antiderivative equals 10 when $x = 3$ ?
- 1
- -1
- 10
- 19
Reveal answer
1 —
Substitute $x = 3$ and antiderivative = 10: $10 = 3^{2} + C = 9 + C$ , so $C = 10 - 9 = - 1$ .

4. Common Pitfalls

Wrong move:

Forgetting to add the constant of integration $+ C$ to an indefinite integral answer.

Why:

Examiners almost always assign 1 mark for $C$ , this is an easy avoidable error.

Correct move:

Always add $+ C$ to the end of every indefinite integral answer.

Wrong move:

Forgetting the $\frac{1}{k}$ scaling factor when integrating $sin (k x), e^{k x}$ and other composite standard functions.

Why:

The scaling factor reverses when moving from differentiation to integration, it is easy to mix up.

Correct move:

Always check your answer by differentiating it to confirm you get the original function back.

Wrong move:

Writing $ln x$ instead of $ln ∣ x ∣$ for the integral of $\frac{1}{x}$ . Missing the absolute value.

Why:

$ln x$ is only defined for positive $x$ , but $\frac{1}{x}$ has an antiderivative for negative $x$ too.

Correct move:

Always include the absolute value sign unless the problem explicitly restricts $x > 0$ .

Wrong move:

Applying the power rule $\frac{x ^{n + 1}}{n + 1}$ to integrate $x^{- 1} = \frac{1}{x}$ .

Why:

The power rule does not work when $n = - 1$ , because division by zero is undefined.

Correct move:

Remember that $\int \frac{1}{x} d x = ln ∣ x ∣ + C$ , this is a special case.

5. Quick Reference Cheatsheet

$f (x)$	$\int f (x) d x$
$x^{n} (n \neq = - 1)$	$\frac{x ^{n + 1}}{n + 1} + C$
$\frac{1}{x}$	$ln ∣ x ∣ + C$
$e^{k x}$	$\frac{1}{k} e^{k x} + C$
$sin (k x)$	$- \frac{1}{k} cos (k x) + C$
$cos (k x)$	$\frac{1}{k} sin (k x) + C$
General rule: If $F^{'} (x) = f (x)$	$\int f (x) d x = F (x) + C$

When this came up on past exams

AI-estimated based on syllabus patterns — cross-check with official past papers for accuracy. Use only as revision-focus signals.

2021 · 1
Find antiderivative with constant
2022 · 2
Applied rate problem with integration
2023 · 1
Antiderivative of trigonometric function

What's Next

Mastering antiderivatives and indefinite integration is the foundation for all further integration topics in IB AI HL. You will use these core skills to solve definite integrals, which let you calculate areas under curves and accumulate quantities over time. Indefinite integration is also essential for solving differential equations, which are widely used to model real-world growth, decay, and motion problems that feature heavily in application-focused AI HL exam questions. Building fluency with basic antiderivative rules now will make more advanced integration techniques much easier to master.