Integration — Reverse of Differentiation and Standard Integrals

CIE IGCSE Additional Mathematics· 14.10, 14.11, 14.12· 25 min read

1. Integration as the Reverse of Differentiation★★☆☆☆⏱ 5 min

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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.

📘 Definition

Indefinite Integral

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

Example:

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

📐 Worked Example

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

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.

2. Integrating Powers of x and 1/(ax+b)★★★☆☆⏱ 7 min

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The core power rule for integration is $\int x^{n} d x = \frac{x ^{n + 1}}{n + 1} + c$ for $n \neq = - 1$ . For $n = - 1$ , use the log rule: $\int \frac{1}{x} d x = ln ∣ x ∣ + c$ . For linear inner terms $a x + b$ , 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 $\int (3 x^{2} + \frac{1}{2 x - 5} + (4 x + 1)^{3}) d x$ .

1
Integrate each term separately, then combine all constants into a single $c$ .
2
Integrate $3 x^{2}$ using the power rule:
3
$\int 3x^2 dx = 3 \times \frac{x^3}{3} = x^3$
4
Integrate $\frac{1}{2 x - 5}$ using the linear log rule:
5
$\int \frac{1}{2x-5} dx = \frac{1}{2} \ln|2x - 5|$
6
Integrate $(4 x + 1)^{3}$ 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 $1/ (a x + b)$ : this is a frequent cause of mark loss in 0606 exams.

3. Integrating Exponential Functions $e^{ax+b}$★★★☆☆⏱ 4 min

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Since $\frac{d}{d x} (e^{a x + b}) = a e^{a x + b}$ , 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 $\int (2 e^{3 x + 2} - e^{1 - 4 x}) d x$ .

1
Integrate each exponential term separately, dividing by the coefficient of $x$ in the exponent each time.
2
Integrate $2 e^{3 x + 2}$ :
3
$\int 2e^{3x+2} dx = 2 \times \frac{1}{3}e^{3x+2} = \frac{2}{3}e^{3x+2}$
4
Integrate $- e^{1 - 4 x}$ , 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
$\frac{2}{3}e^{3x+2} + \frac{1}{4}e^{1-4x} + c$

4. Integrating Trigonometric Functions with Linear Inner Terms★★★★☆⏱ 6 min

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The standard trigonometric integral rules for linear inner terms $a x + b$ 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 $\int (3 cos (2 x - \frac{π}{3}) + 2 sec^{2} (1 - 4 x)) d x$ .

1
Integrate each trigonometric term separately, applying the rules above.
2
Integrate $3 cos (2 x - \frac{π}{3})$ :
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 $2 sec^{2} (1 - 4 x)$ , 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 $sin (a x + b)$ or functions with negative $a$ values: sign errors are the most common mistake in this topic.

5. Common Pitfalls

Wrong move:

Forgetting to add the constant of integration $c$ to indefinite integral answers

Why:

Examiners explicitly award 1 mark for $c$ in nearly all integration questions, so omitting it loses easy marks

Correct move:

Add $c$ to every final answer for indefinite integration problems

Wrong move:

Applying the power rule to $n = - 1$ , writing $\int 1/ x d x = x^{0} /0 + c$

Why:

The power rule only applies for $n \neq = - 1$ , and division by zero is undefined

Correct move:

Use the log rule for $n = - 1$ : $\int 1/ x d x = ln ∣ x ∣ + c$ , and $\int 1/ (a x + b) d x = \frac{1}{a} ln ∣ a x + b ∣ + c$

Wrong move:

Omitting absolute value signs in the log result for integrals of $1/ (a x + b)$

Why:

The logarithm of a negative number is not defined for real numbers, so the absolute value is required

Correct move:

Always write $∣ a x + b ∣$ inside the $ln$ function for these integrals

Wrong move:

Forgetting to divide by the coefficient $a$ when integrating functions with linear inner term $a x + b$

Why:

The chain rule in differentiation multiplies by $a$ , so integration must divide by $a$ to reverse it

Correct move:

Divide by the coefficient of $x$ in the inner linear term for all standard integrals, e.g. $\int sin (3 x) d x = - \frac{c o s ( 3 x )}{3} + c$

Wrong move:

Using the wrong sign for the integral of $sin (a x + b)$ , writing $\int sin (a x + b) d x = \frac{1}{a} cos (a x + b) + c$

Why:

The derivative of $cos (a x + b)$ is $- a sin (a x + b)$ , so reversing the operation introduces a negative sign

Correct move:

Use the rule $\int sin (a x + b) d x = - \frac{1}{a} cos (a x + b) + c$

6. Quick Reference 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 $n = - 1$
$(a x + b)^{n}$ ( $n \neq = - 1$ )	$\frac{( a x + b ) ^{n + 1}}{a ( n + 1 )} + c$	Linear inner power rule
$\frac{1}{a x + b}$	$\frac{1}{a} ln ∣ a x + b ∣ + c$	Linear inner log rule
$e^{a x + b}$	$\frac{1}{a} e^{a x + b} + c$	Exponential integral rule
$sin (a x + b)$	$- \frac{1}{a} cos (a x + b) + c$	Sine integral rule
$cos (a x + b)$	$\frac{1}{a} sin (a x + b) + c$	Cosine integral rule
$sec^{2} (a x + b)$	$\frac{1}{a} tan (a x + b) + c$	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.