Preface 4

History of Integration 8
Beginning of Integration 11

Chapter 1?? Indefinite Integrals 17
Indefinite Integral 18 Formula for (+) in Indefinite Integral 31
Indefinite integral of partial fractions 33 Indefinite integral formula of trigonometric functions 34
Indefinite integral formulas for exponential and logarithmic functions 40
Limit Problems in Indefinite Integrators 42 Indefinite Integrators and Polynomial Functions 53
Substitution Integration Method 55

Chapter 2???? Definite Integrals 61
Section 62: What is a definite integral? 75
The Relationship Between Definite and Indefinite Integrators 78 Properties of Definite Integrators 92
Definite integrals of even and odd functions 92 Definite integrals with absolute values ​​103
Use of substitution integration in definite integrals 113 Use of trigonometric substitution in definite integrals 120
Use of Partial Integration in Definite Integrals 123 Relationship between Infinite Series and Definite Integrals 130

Chapter 3 Area Integral 139
Area between curve and coordinate axis 140 Area between curve and circle 145
Area between curve and line 153

Chapter 4???? Volume Integral 165
Rotate a shape bounded by two curves around an axis 172
Rotate a shape bounded by two curves around an axis 184

Chapter 5???? Velocity, Distance Integration 187
Rectangular Motion 188 Distance in Plane Motion 204

Chapter 6: Uses of Integration in Statistics 209

The power to see the whole by accumulating changes, the beginning of which is integration.
If you want to measure area and volume accurately, it ultimately comes back to integration.
If differentiation shows us moments, integration ties those moments together into a complete picture.
Because integration is one of the most powerful tools that connects mathematics, even if you only know the basic and core formulas for integration, you can apply them to various fields of mathematics.

The moment you understand integration, the structure of functions and graphs begins to become clear at a glance.
We live in a world where things change little by little every day.
How much we walk during the day, how many videos we watch on our smartphones, how much we study, and even how long a cup of coffee lasts on our bodies—these are all 'changes that accumulate little by little' that we cannot immediately see.

Integration is a way to help us understand the big picture by putting these small changes together.
Thinking about difficult formulas and symbols can be daunting, but integration is actually a way of thinking that we already use naturally.
For example, think about increasing experience points in a game.
One small point may not seem like much, but they add up to create a big level up.
We just don't realize it, but this is exactly what integration does.
Integration originated as a formula for calculating area, but in the world we live in today, it is used in a much wider range of fields.
Calculating how many calories we consume in a day, estimating the amount of rainwater we receive during the rainy season, analyzing how caffeine slowly dissipates from our bodies, and even calculating how many hours of video we consume on YouTube each day all require an integral mindset.

In other words, integration is the process of 'dividing a large and complex phenomenon into small pieces and looking at them,' and then reassembling those pieces to understand the whole.
This way of thinking isn't just used in mathematics.
Even today's AI is, in a sense, integrating. When AI understands an image, it doesn't view it as a whole, but rather breaks it down into numerous smaller pieces, analyzes them, and then synthesizes them.
Music recommendation algorithms gather all the little actions we take when we listen—traces like play, skip, and repeat—and use that information to determine our tastes.
Ultimately, all systems work by reading a "collection of small changes," whether predicting risk in finance or detecting signs of disease in medical imaging.
So, integration is not just a formula learned in math class, but rather it is closer to the basic language that moves today's world.
Learning integration is not about memorizing complex symbols, but about developing a sense of seeing the changing world.
We need the ability to understand where a result comes from and what small changes create that result.
The reason why studying doesn't increase, why exercise is less effective, and why time management is difficult all stem from a failure to recognize the "small, accumulating changes."

Once you understand integration, these things begin to seem like a clear structure rather than just a sensation.
When we see how small changes accumulate to create big results, we can make more accurate judgments and wiser choices.
So integration is no longer a skill just for scientists and engineers.
We live in an age where students, office workers, and creators alike must understand the accumulating changes.

Those who understand integration gain the power to read data, broaden their perspective on the passage of time, and gain the ability to calmly break down complex phenomena and reassemble them.
This ability is not for testing, but is a fundamental tool for understanding life.
Ultimately, there is only one reason to learn integration.

The world does not stand still, everything continues to change and accumulate.
The language that tells us how to view and understand this change is integration.
And once we know that language, the movements of the world we live in begin to appear much clearer.
Now, let's build a foundation for understanding the world with "The Easiest Mathematical Integral," which teaches you about the most basic integrals.