# Indefinite integration and anti-differentiation

> IB Mathematics AA SL · IB AA SL Calculus
> Source: https://www.owlsprep.com/study/ib-math-aa-sl-u5-indefinite-integration-and-anti-differentiation/

This sub-topic introduces indefinite integration as the reverse process of differentiation. You will learn to find anti-derivatives of common functions, understand the constant of integration, and solve for particular solutions given initial conditions.

**Prerequisites:** [Basic differentiation of polynomials, trigonometric and exponential functions](https://www.owlsprep.com/study/ib-math-aa-sl-u4-basic-differentiation-rules/)

## Learning objectives

- Recognize the inverse relationship between differentiation and anti-differentiation
- Apply basic integration rules to find indefinite integrals of common functions
- Understand the role of the constant of integration
- Find particular solutions to integrals using initial conditions

## Inverse Relationship Between Differentiation and Integration

**Indefinite Integral** — If $F'(x) = f(x)$ for all $x$ in an interval, then $F(x) + C$ is the indefinite integral of $f(x)$ with respect to $x$. $C$ is the constant of integration.

*Notation:* \int f(x) dx = F(x) + C

*Example:* For $f(x) = 2x$, the indefinite integral is $x^2 + C$

Integration reverses differentiation. Any continuous function has an infinite family of anti-derivatives that differ only by a constant, because the derivative of any constant is zero.

**Worked example:** Find the general indefinite integral of $f(x) = 3x^2 + 4x$

1. Reverse the power rule for differentiation: add 1 to the power of $x$, then divide by the new power, and add the constant of integration.
2. $$\int (3x^2 + 4x) dx = 3 \cdot \frac{x^{2+1}}{2+1} + 4 \cdot \frac{x^{1+1}}{1+1} + C$$
3. Simplify the expression:
4. $$= x^3 + 2x^2 + C$$

> **Exam tip:** Always write $+C$ for every indefinite integral; exam markers will deduct 1 mark if it is missing.

## Standard Integration Rules for Common Functions

| Function $f(x)$ | Indefinite Integral $\int f(x) dx$ |
| --- | --- |
| $x^n \ (n \neq -1)$ | $\frac{x^{n+1}}{n+1} + C$ |
| $\frac{1}{x}$ | $\ln\|x\| + C$ |
| $e^{kx}$ | $\frac{1}{k} e^{kx} + C$ |
| $\sin(kx)$ | $-\frac{1}{k} \cos(kx) + C$ |
| $\cos(kx)$ | $\frac{1}{k} \sin(kx) + C$ |

All these rules are derived by reversing standard differentiation rules. The absolute value in $\ln|x|$ ensures the integral is defined for both positive and negative $x$.

**Worked example:** Find $\int \left(2 \sin 3x + \frac{5}{x} - e^{4x}\right) dx$

1. Integrate each term separately using the standard rules:
2. For $2 \sin 3x$: multiply by $\frac{1}{3}$ and add a negative sign:
3. $$2 \cdot \left(-\frac{1}{3} \cos 3x\right)$$
4. For $\frac{5}{x}$: $5 \ln|x|$, for $-e^{4x}$: multiply by $\frac{1}{4}$:
5. $$+ 5 \ln|x| - \frac{1}{4} e^{4x} + C$$
6. Final simplified result:
7. $$-\frac{2}{3} \cos 3x + 5 \ln|x| - \frac{1}{4} e^{4x} + C$$

*Calculator:* forbidden

## Finding Particular Solutions from Initial Conditions

If you know the value of the original function at a specific point (an initial condition), you can solve for the constant of integration $C$ to get a unique particular solution.

**Worked example:** Given $\frac{dy}{dx} = x^2 + 2$, and $y = 4$ when $x = 1$, find $y$ in terms of $x$.

1. First find the general indefinite integral:
2. $$y = \int (x^2 + 2) dx = \frac{x^3}{3} + 2x + C$$
3. Substitute $y=4$ and $x=1$ to solve for $C$:
4. $$4 = \frac{(1)^3}{3} + 2(1) + C = \frac{7}{3} + C$$
5. Rearrange to find $C$:
6. $$C = 4 - \frac{7}{3} = \frac{5}{3}$$
7. Write the final particular solution:
8. $$y = \frac{1}{3}x^3 + 2x + \frac{5}{3}$$

**Check your understanding**

Test your understanding:

1. If $\frac{dy}{dx} = 2x$ and $y=10$ when $x=3$, what is the value of $C$?

   - 1
   - -1
   - 10
   - 4

   *Answer:* 1

   *Why:* Correct: General solution is $y = x^2 + C$, substitute values: $10 = 3^2 + C \rightarrow C = 1$

*Calculator:* forbidden

## Common pitfalls

- **Wrong:** Forgetting to add the constant of integration $+C$ to an indefinite integral
  - Why it fails: Examiners always penalize this omission, as it is a core requirement of indefinite integration
  - Correct: Write $+C$ at the end of every indefinite integral, even if you will solve for it later
- **Wrong:** Forgetting to divide by the coefficient $k$ when integrating $\sin(kx)$, $\cos(kx)$ or $e^{kx}$
  - Why it fails: This reverses the chain rule from differentiation, which always requires dividing by the constant coefficient
  - Correct: Always multiply the result by $\frac{1}{k}$ when integrating functions of the form $f(kx)$
- **Wrong:** Omitting the absolute value sign when integrating $\frac{1}{x}$
  - Why it fails: $\ln(x)$ is undefined for negative $x$, but $\frac{1}{x}$ is defined for all non-zero $x$
  - Correct: Always write $\ln|x| + C$ when integrating $\frac{1}{x}$
- **Wrong:** Reversing the power rule incorrectly: dividing by the original power instead of the new power
  - Why it fails: This is a common confusion with the differentiation power rule
  - Correct: Remember: add 1 to the power, then divide by the new power
- **Wrong:** Writing multiple constants of integration for multiple terms, e.g. $\frac{x^3}{3} + C + 2x + C$
  - Why it fails: All arbitrary constants can be combined into a single constant $C$
  - Correct: Write only one $+C$ at the end of the entire integral

## Cheatsheet

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

## What's next

Indefinite integration is the foundation for all further integration topics in IB AA SL. Next you will learn definite integration and the Fundamental Theorem of Calculus, which connects indefinite integrals to calculating areas under curves. From there, you will move on to integration by substitution and applications of integration to kinematics and area problems. Mastering the basic rules of indefinite integration here will make all these more advanced topics much simpler to grasp.

- [Definite integration and fundamental theorem of calculus](https://www.owlsprep.com/study/ib-math-aa-sl-u5-definite-integration-and-fundamental-theorem/)
- [Integration by substitution, area and volume of revolution](https://www.owlsprep.com/study/ib-math-aa-sl-u5-integration-by-substitution-area-and/)

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