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:
$$
\int 3x^2 dx
$$

This course distills the essentials of Python's math module, focusing on trigonometric functions, logarithmic operations, and mathematical constants. Learners will gain hands-on experience using Python to perform these standard mathematical computations, ideal for beginners seeking practical skills with the math module.

Introduction to Integrals

Basic Integral

Common Integral Rules

Power Rule for Integration

Exponential Rule

Trigonometric Integrals

Definite Integrals