Summary  
This chapter explains how to find antiderivatives using the power, exponential, and trigonometric integration rules and how to evaluate definite integrals to compute the area under a curve.

General domain of usage  
Scientific computing

**Integration** is a fundamental concept in calculus that represents the total accumulation of a quantity, such as the area under a curve. It is essential in data science for calculating probability distributions, cumulative values, and optimization.


Definition

## Basic Integral
The basic integral of a power function follows this rule:

$$
\int Cx^ndx = C\left( \frac{x^{n+1}}{n+1} \right) + C
$$
 
Where:
- $$C$$ is a constant;
- $$n \neq -1$$;
- $$...+C$$ represents an arbitrary constant of integration.

**Key idea**: if differentiation reduces the power of $$x$$, integration increases it.


## Common Integral Rules

### Power Rule for Integration
This rule helps integrate any polynomial expression:

$$
\int x^ndx = \frac{x^{n+1}}{n+1}+ C,\ n \neq -1
$$

For example, if $$n = 2$$:
$$
\int x^2dx = \frac{x^3}{3}+C
$$
 


### Exponential Rule
The integral of the exponential function $$e^x$$ is unique because it remains the same after integration:

$$
\int e^xdx = e^x + C
$$

But if we have an exponent with a coefficient, we use another rule:

$$
\int e^{ax}dx = \frac{1}{a}e^{ax}+C,\ a \neq 0
$$

For example, if $$a = 2$$:

$$
\int e^{2x}dx = \frac{e^{2x}}{2} + C
$$
 


### Trigonometric Integrals
Sine and cosine functions also follow straightforward integration rules:

$$
\int sin(x)dx = -cos(x) + C \\ \int cos(x)dx = sin(x) + C
$$
 


## Definite Integrals

Unlike indefinite integrals, which include an arbitrary constant $$C$$, definite integrals evaluate a function **between two limits** $$a$$ and $$b$$:

$$
\int_a^b f(x)dx = F(b) - F(a)
$$

Where $$F(x)$$ is the **antiderivative** of $$f(x)$$.

For example, if $$f(x) = 2x$$, $$a = 0$$ and $$b = 2$$:

$$
\int^2_0 2x\ dx = \left[ x^2 \right] = 4 - 0 = 4
$$
 
This means the **area under the curve** $$y = 2x$$ from $$x=0$$ to $$x=2$$ is $$4$$.


Calculate the integral:
$$
\int 3x^2 dx
$$

Master the mathematical foundations essential for data science. Explore core concepts in functions, calculus, linear algebra, probability, and dimensionality reduction. Build both theoretical understanding and practical coding experience to strengthen your ability to analyze data, model complex systems, and apply advanced techniques in machine learning.

Explore the foundation of mathematical functions. Learn different types of algebraic and transcendental functions, their properties, and how to implement them in Python to solve real-world problems.

Master the concepts of sets and series, from basic operations to practical applications. Gain hands-on experience implementing set operations and working with arithmetic and geometric series in Python.

Develop a solid understanding of limits, derivatives, integrals, and partial derivatives. Connect theory to practice by implementing these concepts in Python and applying them to optimization through gradient descent.

Build strong knowledge of vectors, matrices, and transformations. Learn decomposition methods and eigenvalue analysis, while reinforcing concepts with Python coding challenges and practical data science applications.

Dive into probability theory and statistics. Study conditional probability, Bayes' theorem, and statistical measures. Implement key concepts in Python, simulate distributions, and strengthen your skills through challenges and quizzes.

Introduction to Integrals

Basic Integral

Common Integral Rules

Power Rule for Integration

Exponential Rule

Trigonometric Integrals

Definite Integrals