MASTERING INTEGRATION
From Basics to Advanced
A Comprehensive Workbook with 200+ Solved Problems
Perfect for:
SAT • ACT • AP Calculus • IB • A-Levels
University Entrance Exams
y
Area = e − 1
x
Author: Rachid Ousalem
March 24, 2026
, Copyright © 2024 Rachid Ousalem
All rights reserved. No part of this book may be reproduced
without written permission from the author.
,Contents
Introduction 5
1 Integration: Complete Summary 7
2 Basic Exercises: Building Foundations 11
Solutions to Basic Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15
3 Intermediate Exercises: Building Skills 19
4 Advanced Exercises: Mastery Level 23
5 Geometric Applications: Area, Volume Length 27
6 Practice Exams: Test Your Mastery 31
7 Complete Solutions 35
Appendix: Formula Sheet 38
8 Intermediate Exercises: Building Skills 41
Scientific Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42
9 Advanced Exercises: Mastery Level 49
10 Geometric Applications: Area Between Curves 55
Additional Graphs Gallery . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 67
3
,4/73
,Introduction
Welcome to Integration!
Integration is one of the most powerful tools in mathematics. It allows us to:
• Calculate areas under curves
• Find volumes of solids
• Determine total accumulation
• Solve differential equations
This workbook will take you from the fundamentals to advanced integration tech-
niques through carefully designed problems and detailed solutions.
How to Use This Book
1. Study the Summary - Review the key formulas and concepts
2. Follow the Examples - Work through solved problems step by step
3. Practice with Exercises - Start with basic, then progress to advanced
4. Check Solutions - Use the detailed solutions to verify your work
5. Take the Practice Exams - Test your mastery
Topics Covered
• Basic Integration Rules • Partial Fractions
• Integration by Substitution
• Area Between Curves
• Integration by Parts
• Volume of Revolution
• Trigonometric Integrals
• Trigonometric Substitution • Improper Integrals
5
,Color Code
• Blue: Basic Level
• Orange: Intermediate Level
• Red: Advanced Level
• Gold: Formulas and Key Concepts
6/73
,Chapter 1
Integration: Complete Summary
Table of Basic Integrals
Function
Z Integral
k dx kx + C
xn+1
Z
xn dx +C (n ̸= −1)
Z n+1
1
dx ln |x| + C
Z x
ex dx ex + C
ax
Z
ax dx +C
Z ln a
sin x dx − cos x + C
Z
cos x dx sin x + C
Z
sec2 x dx tan x + C
Z
csc2 x dx − cot x + C
Z
sec x tan x dx sec x + C
Z
csc x cot x dx − csc x + C
Z
1
dx arctan x + C
Z 1 + x2
1
√ dx arcsin x + C
1 − x2
Golden Rule: Always add +C for indefinite integrals!
7
,Integration Techniques
1. Substitution Method
If u = g(x) and du = g ′ (x)dx, then:
Z Z
′
f (g(x))g (x) dx = f (u) du
Look for a function and its derivative inside the integral.
2. Integration by Parts
Z Z
u dv = uv − v du
Choose u using LIATE order:
• Logarithmic
• Inverse trigonometric
• Algebraic
• Trigonometric
• Exponential
3. Trigonometric Integrals
Z
x sin 2x
sin2 x dx =− +C
2 4
Z
x sin 2x
cos2 x dx = + +C
2 4
Z
tan2 x dx = tan x − x + C
4. Trigonometric Substitution
Expression
√ Substitution Identity
2
√a − x
2 x = a sin θ 1 − sin2 θ = cos2 θ
2
√a + x
2 x = a tan θ 1 + tan2 θ = sec2 θ
x2 − a 2 x = a sec θ sec2 θ − 1 = tan2 θ
5. Partial Fractions
For rational functions P (x)
Q(x)
:
• Factor Q(x)
• Decompose into simpler fractions
• Integrate each term
8/73
,Definite Integrals and Applications
Fundamental Theorem of Calculus
Z b
f (x) dx = F (b) − F (a)
a
where F is an antiderivative of f .
Area Between Curves
Z b
Area = [f (x) − g(x)] dx
a
where f (x) ≥ g(x) on [a, b].
Volume of Revolution
Washer Method (around x-axis):
Z b
V =π [R(x)2 − r(x)2 ] dx
a
Shell Method: Z b
V = 2π xf (x) dx
a
Average Value of a Function
Z b
1
favg = f (x) dx
b−a a
Length of a Curve
Z bp
L= 1 + [f ′ (x)]2 dx
a
9/73
, Quick Reference Card
The Most Important Integrals to Memorize
Z
xn+1
Z
• n
x dx = • cos xdx = sin x
n+1
Z Z
1
• dx = ln |x| • sec2 xdx = tan x
x
Z
1
Z
• ex dx = ex • dx = arctan x
1 + x2
Z
1
Z
• sin xdx = − cos x • √ dx = arcsin x
1 − x2
[Formula] Formula:Euler’s Formula (for the curious):
eiθ = cos θ + i sin θ
[Tip] Pro Tip:When in doubt, differentiate your answer to check!
[Warning] Warning:Common Mistakes:
• Forgetting the +C for indefinite integrals
• Confusing x1 dx with x−1 dx (they’re the same!)
R R
• Forgetting the chain rule in substitution
10/73
From Basics to Advanced
A Comprehensive Workbook with 200+ Solved Problems
Perfect for:
SAT • ACT • AP Calculus • IB • A-Levels
University Entrance Exams
y
Area = e − 1
x
Author: Rachid Ousalem
March 24, 2026
, Copyright © 2024 Rachid Ousalem
All rights reserved. No part of this book may be reproduced
without written permission from the author.
,Contents
Introduction 5
1 Integration: Complete Summary 7
2 Basic Exercises: Building Foundations 11
Solutions to Basic Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15
3 Intermediate Exercises: Building Skills 19
4 Advanced Exercises: Mastery Level 23
5 Geometric Applications: Area, Volume Length 27
6 Practice Exams: Test Your Mastery 31
7 Complete Solutions 35
Appendix: Formula Sheet 38
8 Intermediate Exercises: Building Skills 41
Scientific Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42
9 Advanced Exercises: Mastery Level 49
10 Geometric Applications: Area Between Curves 55
Additional Graphs Gallery . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 67
3
,4/73
,Introduction
Welcome to Integration!
Integration is one of the most powerful tools in mathematics. It allows us to:
• Calculate areas under curves
• Find volumes of solids
• Determine total accumulation
• Solve differential equations
This workbook will take you from the fundamentals to advanced integration tech-
niques through carefully designed problems and detailed solutions.
How to Use This Book
1. Study the Summary - Review the key formulas and concepts
2. Follow the Examples - Work through solved problems step by step
3. Practice with Exercises - Start with basic, then progress to advanced
4. Check Solutions - Use the detailed solutions to verify your work
5. Take the Practice Exams - Test your mastery
Topics Covered
• Basic Integration Rules • Partial Fractions
• Integration by Substitution
• Area Between Curves
• Integration by Parts
• Volume of Revolution
• Trigonometric Integrals
• Trigonometric Substitution • Improper Integrals
5
,Color Code
• Blue: Basic Level
• Orange: Intermediate Level
• Red: Advanced Level
• Gold: Formulas and Key Concepts
6/73
,Chapter 1
Integration: Complete Summary
Table of Basic Integrals
Function
Z Integral
k dx kx + C
xn+1
Z
xn dx +C (n ̸= −1)
Z n+1
1
dx ln |x| + C
Z x
ex dx ex + C
ax
Z
ax dx +C
Z ln a
sin x dx − cos x + C
Z
cos x dx sin x + C
Z
sec2 x dx tan x + C
Z
csc2 x dx − cot x + C
Z
sec x tan x dx sec x + C
Z
csc x cot x dx − csc x + C
Z
1
dx arctan x + C
Z 1 + x2
1
√ dx arcsin x + C
1 − x2
Golden Rule: Always add +C for indefinite integrals!
7
,Integration Techniques
1. Substitution Method
If u = g(x) and du = g ′ (x)dx, then:
Z Z
′
f (g(x))g (x) dx = f (u) du
Look for a function and its derivative inside the integral.
2. Integration by Parts
Z Z
u dv = uv − v du
Choose u using LIATE order:
• Logarithmic
• Inverse trigonometric
• Algebraic
• Trigonometric
• Exponential
3. Trigonometric Integrals
Z
x sin 2x
sin2 x dx =− +C
2 4
Z
x sin 2x
cos2 x dx = + +C
2 4
Z
tan2 x dx = tan x − x + C
4. Trigonometric Substitution
Expression
√ Substitution Identity
2
√a − x
2 x = a sin θ 1 − sin2 θ = cos2 θ
2
√a + x
2 x = a tan θ 1 + tan2 θ = sec2 θ
x2 − a 2 x = a sec θ sec2 θ − 1 = tan2 θ
5. Partial Fractions
For rational functions P (x)
Q(x)
:
• Factor Q(x)
• Decompose into simpler fractions
• Integrate each term
8/73
,Definite Integrals and Applications
Fundamental Theorem of Calculus
Z b
f (x) dx = F (b) − F (a)
a
where F is an antiderivative of f .
Area Between Curves
Z b
Area = [f (x) − g(x)] dx
a
where f (x) ≥ g(x) on [a, b].
Volume of Revolution
Washer Method (around x-axis):
Z b
V =π [R(x)2 − r(x)2 ] dx
a
Shell Method: Z b
V = 2π xf (x) dx
a
Average Value of a Function
Z b
1
favg = f (x) dx
b−a a
Length of a Curve
Z bp
L= 1 + [f ′ (x)]2 dx
a
9/73
, Quick Reference Card
The Most Important Integrals to Memorize
Z
xn+1
Z
• n
x dx = • cos xdx = sin x
n+1
Z Z
1
• dx = ln |x| • sec2 xdx = tan x
x
Z
1
Z
• ex dx = ex • dx = arctan x
1 + x2
Z
1
Z
• sin xdx = − cos x • √ dx = arcsin x
1 − x2
[Formula] Formula:Euler’s Formula (for the curious):
eiθ = cos θ + i sin θ
[Tip] Pro Tip:When in doubt, differentiate your answer to check!
[Warning] Warning:Common Mistakes:
• Forgetting the +C for indefinite integrals
• Confusing x1 dx with x−1 dx (they’re the same!)
R R
• Forgetting the chain rule in substitution
10/73
