FURTHER MATHEMATICS
INTEGRATION
Complete Examination & Revision Study Note
Level: Senior Secondary / WAEC-style Further Mathematics
Topic: Integration
Purpose: Learning, revision, and examination preparation
A complete, self-contained reference covering indefinite and definite
integration, standard rules, substitution, integration by parts, areas,
and applications — with worked examples, practice questions, and a full answer key.
Study Notes Series — Companion to the Differentiation Study Note
,Further Mathematics — Integration Study & Revision Notes
Contents
1 Introduction to Integration 2
2 The Power Rule for Integration 2
3 Integrating Constants 2
4 Constant Multiplier Rule 3
5 Sum and Difference Rule 3
6 Integrating Fractions and Roots 3
7 Integrating Exponential Functions 4
8 Integrating Trigonometric Functions 4
9 Integration Involving the Chain Rule (Linear Substitution) 4
10 Integration by Substitution 4
11 Integration by Parts 5
12 The Definite Integral 5
13 Area Under a Curve 6
14 Area Between Two Curves 6
15 Applications of Integration 6
16 Integration and Motion 7
17 Common Integration Mistakes 7
18 Quick Integration Table 8
19 Exam-Ready Checklist 8
20 Practice Questions 9
21 Answer Key 10
One-Page Memory Summary 11
,Further Mathematics — Integration Study & Revision Notes
1 Introduction to Integration
Integration is the reverse process of differentiation. If differentiation finds the rate of change of a
function, integration finds the original function from its rate of change — it is often called antid-
ifferentiation.
Z
d
If [F (x)] = f (x), then f (x) dx = F (x) + C
dx
The constant C is called the constant of integration. It appears because the derivative of
any constant is zero, so infinitely many functions share the same derivative.
Worked Example
Z
d 2
If (x ) = 2x, then 2x dx = x2 + C.
dx
2 The Power Rule for Integration
xn+1
Z
xn dx = +C (n ̸= −1)
n+1
Add 1 to the power, then divide by the new power.
Worked Example
x4
Z
1. x3 dx = +C
4
Z
x 7
2. x6 dx = +C
Z7
x−1
Z
1 −2 1
3. 2
dx = x dx = +C =− +C
x −1 x
Important exception: The power rule does not work when n = −1, since dividing by zero
is undefined. Instead:
Z
1
dx = ln |x| + C
x
3 Integrating Constants
Z
k dx = kx + C
Worked Example
Z
7 dx = 7x + C
2
, Further Mathematics — Integration Study & Revision Notes
4 Constant Multiplier Rule
Z Z
k f (x) dx = k f (x) dx
Worked Example
x4 5x4
Z Z
3
5x dx = 5 x3 dx = 5 · +C = +C
4 4
5 Sum and Difference Rule
Z Z Z
[f (x) ± g(x)] dx = f (x) dx ± g(x) dx
Worked Example
Integrate y = 6x2 − 4x + 5.
Z
(6x2 − 4x + 5) dx = 2x3 − 2x2 + 5x + C
Exam Tip: Always add + C for an indefinite integral — omitting it is one of the most
commonly penalized mistakes.
6 Integrating Fractions and Roots
Rewrite using negative or fractional indices before integrating.
Worked Example
4
y= = 4x−3
x3
x−2
Z
2
4x−3 dx = 4 · +C =− 2 +C
−2 x
Worked Example
√
y= x = x1/2
x3/2
Z
2
x1/2 dx = + C = x3/2 + C
3/2 3
3
INTEGRATION
Complete Examination & Revision Study Note
Level: Senior Secondary / WAEC-style Further Mathematics
Topic: Integration
Purpose: Learning, revision, and examination preparation
A complete, self-contained reference covering indefinite and definite
integration, standard rules, substitution, integration by parts, areas,
and applications — with worked examples, practice questions, and a full answer key.
Study Notes Series — Companion to the Differentiation Study Note
,Further Mathematics — Integration Study & Revision Notes
Contents
1 Introduction to Integration 2
2 The Power Rule for Integration 2
3 Integrating Constants 2
4 Constant Multiplier Rule 3
5 Sum and Difference Rule 3
6 Integrating Fractions and Roots 3
7 Integrating Exponential Functions 4
8 Integrating Trigonometric Functions 4
9 Integration Involving the Chain Rule (Linear Substitution) 4
10 Integration by Substitution 4
11 Integration by Parts 5
12 The Definite Integral 5
13 Area Under a Curve 6
14 Area Between Two Curves 6
15 Applications of Integration 6
16 Integration and Motion 7
17 Common Integration Mistakes 7
18 Quick Integration Table 8
19 Exam-Ready Checklist 8
20 Practice Questions 9
21 Answer Key 10
One-Page Memory Summary 11
,Further Mathematics — Integration Study & Revision Notes
1 Introduction to Integration
Integration is the reverse process of differentiation. If differentiation finds the rate of change of a
function, integration finds the original function from its rate of change — it is often called antid-
ifferentiation.
Z
d
If [F (x)] = f (x), then f (x) dx = F (x) + C
dx
The constant C is called the constant of integration. It appears because the derivative of
any constant is zero, so infinitely many functions share the same derivative.
Worked Example
Z
d 2
If (x ) = 2x, then 2x dx = x2 + C.
dx
2 The Power Rule for Integration
xn+1
Z
xn dx = +C (n ̸= −1)
n+1
Add 1 to the power, then divide by the new power.
Worked Example
x4
Z
1. x3 dx = +C
4
Z
x 7
2. x6 dx = +C
Z7
x−1
Z
1 −2 1
3. 2
dx = x dx = +C =− +C
x −1 x
Important exception: The power rule does not work when n = −1, since dividing by zero
is undefined. Instead:
Z
1
dx = ln |x| + C
x
3 Integrating Constants
Z
k dx = kx + C
Worked Example
Z
7 dx = 7x + C
2
, Further Mathematics — Integration Study & Revision Notes
4 Constant Multiplier Rule
Z Z
k f (x) dx = k f (x) dx
Worked Example
x4 5x4
Z Z
3
5x dx = 5 x3 dx = 5 · +C = +C
4 4
5 Sum and Difference Rule
Z Z Z
[f (x) ± g(x)] dx = f (x) dx ± g(x) dx
Worked Example
Integrate y = 6x2 − 4x + 5.
Z
(6x2 − 4x + 5) dx = 2x3 − 2x2 + 5x + C
Exam Tip: Always add + C for an indefinite integral — omitting it is one of the most
commonly penalized mistakes.
6 Integrating Fractions and Roots
Rewrite using negative or fractional indices before integrating.
Worked Example
4
y= = 4x−3
x3
x−2
Z
2
4x−3 dx = 4 · +C =− 2 +C
−2 x
Worked Example
√
y= x = x1/2
x3/2
Z
2
x1/2 dx = + C = x3/2 + C
3/2 3
3
