# Antiderivatives and indefinite integration

> IB Mathematics Applications and Interpretation HL · IB AI HL Calculus
> Source: https://www.owlsprep.com/study/ib-math-ai-hl-u4-antiderivatives-and-indefinite-integration/

This sub-topic introduces indefinite integration as the inverse of differentiation, covers rules for finding antiderivatives of standard functions, and explains how to find the constant of integration for initial value problems.

**Prerequisites:** [Basic differentiation of standard functions](https://www.owlsprep.com/study/ib-math-ai-hl-u4-differentiation-of-standard-functions/)

## Learning objectives

- Recognize differentiation and integration as inverse operations
- Calculate antiderivatives of common standard functions
- Find the constant of integration from initial conditions
- Set up indefinite integrals for basic applied rate problems

## Antiderivatives as the inverse of differentiation

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**.

**Antiderivative** — A function $F(x)$ is an antiderivative of $f(x)$ on an interval if $F'(x) = f(x)$ for all $x$ in that interval.

*Notation:* $F(x) + C, C \in \mathbb{R}$

**Worked example:** Verify that $F(x) = x^3 + 2x + 5$ is an antiderivative of $f(x) = 3x^2 + 2$.

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 $C$.

> **tip**
>
> You can always check your integration answer by differentiating it — if you get back the original function, your antiderivative is correct.

> **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.

## Standard integration rules

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 $\int f(x) dx$ |
| --- | --- |
| $x^n \ (n \neq -1)$ | $\frac{x^{n+1}}{n+1} + C$ |
| $\frac{1}{x} = x^{-1}$ | $\ln\|x\| + C$ |
| $e^{kx}$ | $\frac{1}{k}e^{kx} + C$ |
| $\sin(kx)$ | $-\frac{1}{k}\cos(kx) + C$ |
| $\cos(kx)$ | $\frac{1}{k}\sin(kx) + C$ |
| $\sec^2(kx)$ | $\frac{1}{k}\tan(kx) + C$ |

**Worked example:** Calculate $\int \left(4x^3 - 2\sin 2x + \frac{3}{x}\right) dx$.

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. $$\int \left(4x^3 - 2\sin 2x + \frac{3}{x}\right) dx = x^4 + \cos 2x + 3\ln|x| + C$$

> **warning**
>
> Always remember the scaling factor $\frac{1}{k}$ for composite standard functions like $\sin(kx)$. It is easy to mix this up with the scaling factor from differentiation.

## Finding the constant of integration

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.

**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 $\frac{dy}{dx} = 3x^2 - 2x$, 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. $$y = x^3 - x^2$$

**Check your understanding**

Test your understanding:

1. What is the value of $C$ if $\int 2x dx = x^2 + C$ and the antiderivative equals 10 when $x=3$?

   - 1
   - -1
   - 10
   - 19

   *Answer:* -1

   *Why:* Substitute $x=3$ and antiderivative = 10: $10 = 3^2 + C = 9 + C$, so $C = 10 - 9 = -1$.

## Common pitfalls

- **Wrong:** Forgetting to add the constant of integration $+C$ to an indefinite integral answer.
  - Why it fails: Examiners almost always assign 1 mark for $C$, this is an easy avoidable error.
  - Correct: Always add $+C$ to the end of every indefinite integral answer.
- **Wrong:** Forgetting the $\frac{1}{k}$ scaling factor when integrating $\sin(kx), e^{kx}$ and other composite standard functions.
  - Why it fails: The scaling factor reverses when moving from differentiation to integration, it is easy to mix up.
  - Correct: Always check your answer by differentiating it to confirm you get the original function back.
- **Wrong:** Writing $\ln x$ instead of $\ln|x|$ for the integral of $\frac{1}{x}$. Missing the absolute value.
  - Why it fails: $\ln x$ is only defined for positive $x$, but $\frac{1}{x}$ has an antiderivative for negative $x$ too.
  - Correct: Always include the absolute value sign unless the problem explicitly restricts $x > 0$.
- **Wrong:** Applying the power rule $\frac{x^{n+1}}{n+1}$ to integrate $x^{-1} = \frac{1}{x}$.
  - Why it fails: The power rule does not work when $n=-1$, because division by zero is undefined.
  - Correct: Remember that $\int \frac{1}{x} dx = \ln|x| + C$, this is a special case.

## Cheatsheet

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

## 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.

- [Definite integrals and area under curves](https://www.owlsprep.com/study/ib-math-ai-hl-u4-definite-integrals-and-area-under/)
- [Integration by substitution](https://www.owlsprep.com/study/ib-math-ai-hl-u4-integration-techniques-substitution-and-by/)
- [Area between curves, volumes of revolution](https://www.owlsprep.com/study/ib-math-ai-hl-u4-area-between-curves-volumes-of/)

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