# Integration — Reverse of Differentiation and Standard Integrals

> CIE IGCSE Additional Mathematics · 0606 (2025-2027)
> Source: https://www.owlsprep.com/study/cie-0606-u14-integration-reverse-of-differentiation-and/

This guide explains integration as the reverse of differentiation, introduces the constant of integration, and covers all standard integral formulas required for CIE IGCSE Additional Mathematics 0606, with exam-aligned worked examples.

**Prerequisites:** Differentiation of powers of x, trigonometric, exponential and log functions; Simplification of algebraic and trigonometric expressions

## Learning objectives

- Understand integration as the inverse process of differentiation, including the constant of integration c
- Integrate powers of x including 1/x and 1/(ax+b) for linear ax+b
- Apply standard integral formulas for (ax+b)ⁿ, sin(ax+b), cos(ax+b), sec²(ax+b) and e^(ax+b)
- Solve basic indefinite integration questions aligned to 0606 exam requirements

## Integration as the Reverse of Differentiation

Integration is the inverse operation of differentiation: if the derivative of $F(x)$ is $f(x)$, then the integral of $f(x)$ is $F(x) + c$, where $c$ is the constant of integration. This relationship is the foundation of all integral calculus for the 0606 syllabus.

**Indefinite Integral** — The full set of antiderivatives of a given integrand, where all results differ only by a constant value $c$

*Example:* If $\frac{d}{dx}(x^2) = 2x$, then $\int 2x \, dx = x^2 + c$

**Worked example:** Given that $\frac{d}{dx}(3x^4 - 2x + 5) = 12x^3 - 2$, find $\int (12x^3 - 2) \, dx$.

1. Recognize integration reverses differentiation, so the integral is the original function plus the constant of integration.
2. Write the final result, combining all arbitrary constants into a single $c$:
3. $$\int (12x^3 - 2) dx = 3x^4 - 2x + c$$

> **Exam tip:** Always add the constant of integration $c$ to every indefinite integral answer: it is worth 1 mark in nearly all 0606 integration questions.

*Calculator:* forbidden

## Integrating Powers of x and 1/(ax+b)

The core power rule for integration is $\int x^n dx = \frac{x^{n+1}}{n+1} + c$ for $n \neq -1$. For $n=-1$, use the log rule: $\int \frac{1}{x} dx = \ln|x| + c$. For linear inner terms $ax+b$, extend the rule by dividing by the coefficient $a$ of $x$:

$$\int (ax+b)^n dx = \frac{(ax+b)^{n+1}}{a(n+1)} + c \quad (n \neq -1)$$

$$\int \frac{1}{ax+b} dx = \frac{1}{a} \ln|ax+b| + c$$

**Worked example:** Find $\int \left(3x^2 + \frac{1}{2x - 5} + (4x + 1)^3\right) dx$.

1. Integrate each term separately, then combine all constants into a single $c$.
2. Integrate $3x^2$ using the power rule:
3. $$\int 3x^2 dx = 3 \times \frac{x^3}{3} = x^3$$
4. Integrate $\frac{1}{2x-5}$ using the linear log rule:
5. $$\int \frac{1}{2x-5} dx = \frac{1}{2} \ln|2x - 5|$$
6. Integrate $(4x+1)^3$ using the linear power rule:
7. $$\int (4x+1)^3 dx = \frac{(4x+1)^4}{4 \times 4} = \frac{(4x+1)^4}{16}$$
8. Combine terms and add $c$ for the final result:
9. $$x^3 + \frac{1}{2}\ln|2x - 5| + \frac{(4x+1)^4}{16} + c$$

> **Exam tip:** Never omit the absolute value signs inside the $\ln$ function for integrals of $1/(ax+b)$: this is a frequent cause of mark loss in 0606 exams.

*Calculator:* forbidden

## Integrating Exponential Functions $e^{ax+b}$

Since $\frac{d}{dx}\left(e^{ax+b}\right) = ae^{ax+b}$, reversing the differentiation gives the integral rule for exponential functions with linear inner terms:

$$\int e^{ax+b} dx = \frac{1}{a}e^{ax+b} + c$$

**Worked example:** Evaluate $\int \left(2e^{3x+2} - e^{1-4x}\right) dx$.

1. Integrate each exponential term separately, dividing by the coefficient of $x$ in the exponent each time.
2. Integrate $2e^{3x+2}$:
3. $$\int 2e^{3x+2} dx = 2 \times \frac{1}{3}e^{3x+2} = \frac{2}{3}e^{3x+2}$$
4. Integrate $-e^{1-4x}$, taking care with the negative coefficient of $x$:
5. $$\int -e^{1-4x} dx = - \times \frac{1}{-4}e^{1-4x} = \frac{1}{4}e^{1-4x}$$
6. Combine terms and add $c$:
7. $$\frac{2}{3}e^{3x+2} + \frac{1}{4}e^{1-4x} + c$$

*Calculator:* allowed

## Integrating Trigonometric Functions with Linear Inner Terms

The standard trigonometric integral rules for linear inner terms $ax+b$ are derived directly from reversing differentiation rules, taking care to include correct signs and divide by $a$:

$$\int \sin(ax+b) dx = -\frac{1}{a}\cos(ax+b) + c$$

$$\int \cos(ax+b) dx = \frac{1}{a}\sin(ax+b) + c$$

$$\int \sec^2(ax+b) dx = \frac{1}{a}\tan(ax+b) + c$$

**Worked example:** Find $\int \left(3\cos\left(2x - \frac{\pi}{3}\right) + 2\sec^2(1 - 4x)\right) dx$.

1. Integrate each trigonometric term separately, applying the rules above.
2. Integrate $3\cos\left(2x - \frac{\pi}{3}\right)$:
3. $$\int 3\cos\left(2x - \frac{\pi}{3}\right) dx = 3 \times \frac{1}{2}\sin\left(2x - \frac{\pi}{3}\right) = \frac{3}{2}\sin\left(2x - \frac{\pi}{3}\right)$$
4. Integrate $2\sec^2(1-4x)$, taking care with the negative coefficient of $x$:
5. $$\int 2\sec^2(1-4x) dx = 2 \times \frac{1}{-4}\tan(1-4x) = -\frac{1}{2}\tan(1-4x)$$
6. Combine terms and add $c$ for the final result:
7. $$\frac{3}{2}\sin\left(2x - \frac{\pi}{3}\right) - \frac{1}{2}\tan(1-4x) + c$$

> **Exam tip:** Double-check the sign of your result when integrating $\sin(ax+b)$ or functions with negative $a$ values: sign errors are the most common mistake in this topic.

*Calculator:* allowed

## Common pitfalls

- **Wrong:** Forgetting to add the constant of integration $c$ to indefinite integral answers
  - Why it fails: Examiners explicitly award 1 mark for $c$ in nearly all integration questions, so omitting it loses easy marks
  - Correct: Add $c$ to every final answer for indefinite integration problems
- **Wrong:** Applying the power rule to $n=-1$, writing $\int 1/x dx = x^0/0 + c$
  - Why it fails: The power rule only applies for $n \neq -1$, and division by zero is undefined
  - Correct: Use the log rule for $n=-1$: $\int 1/x dx = \ln|x| + c$, and $\int 1/(ax+b) dx = \frac{1}{a}\ln|ax+b| + c$
- **Wrong:** Omitting absolute value signs in the log result for integrals of $1/(ax+b)$
  - Why it fails: The logarithm of a negative number is not defined for real numbers, so the absolute value is required
  - Correct: Always write $|ax+b|$ inside the $\ln$ function for these integrals
- **Wrong:** Forgetting to divide by the coefficient $a$ when integrating functions with linear inner term $ax+b$
  - Why it fails: The chain rule in differentiation multiplies by $a$, so integration must divide by $a$ to reverse it
  - Correct: Divide by the coefficient of $x$ in the inner linear term for all standard integrals, e.g. $\int \sin(3x) dx = -\frac{\cos(3x)}{3} + c$
- **Wrong:** Using the wrong sign for the integral of $\sin(ax+b)$, writing $\int \sin(ax+b) dx = \frac{1}{a}\cos(ax+b) + c$
  - Why it fails: The derivative of $\cos(ax+b)$ is $-a\sin(ax+b)$, so reversing the operation introduces a negative sign
  - Correct: Use the rule $\int \sin(ax+b) dx = -\frac{1}{a}\cos(ax+b) + c$

## Cheatsheet

| Integrand | Integral Result | Key Note |
| --- | --- | --- |
| $x^n$ ($n \neq -1$) | $\frac{x^{n+1}}{n+1} + c$ | Basic power rule |
| $\frac{1}{x}$ | $\ln\|x\| + c$ | Log rule for $n=-1$ |
| $(ax+b)^n$ ($n \neq -1$) | $\frac{(ax+b)^{n+1}}{a(n+1)} + c$ | Linear inner power rule |
| $\frac{1}{ax+b}$ | $\frac{1}{a}\ln\|ax+b\| + c$ | Linear inner log rule |
| $e^{ax+b}$ | $\frac{1}{a}e^{ax+b} + c$ | Exponential integral rule |
| $\sin(ax+b)$ | $-\frac{1}{a}\cos(ax+b) + c$ | Sine integral rule |
| $\cos(ax+b)$ | $\frac{1}{a}\sin(ax+b) + c$ | Cosine integral rule |
| $\sec^2(ax+b)$ | $\frac{1}{a}\tan(ax+b) + c$ | Secant squared integral rule |

## What's next

Now that you have mastered the standard integral rules for CIE IGCSE Additional Mathematics 0606, you are ready to progress to more advanced integration topics. Next, you will learn to evaluate definite integrals, use integration to calculate the area under a curve, and solve basic first-order differential equations, all of which are heavily tested in both Paper 1 (non-calculator) and Paper 2 (calculator) of the exam. Practicing past paper questions on standard integrals will help you eliminate common sign errors and remember to include the constant of integration, ensuring you pick up all available marks for this high-frequency topic.

---

From [OwlsPrep](https://www.owlsprep.com) — free study guides for A-Level, IB, AP and IGCSE, written against the official syllabus. Canonical page: https://www.owlsprep.com/study/cie-0606-u14-integration-reverse-of-differentiation-and/