Learn Introduction to Integrals | Mathematical Analysis

Definition

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.

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 $4$ .

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Section 3. Chapter 5

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