Integration is a fundamental concept in calculus that helps us compute the total accumulation of a quantity, such as areas under curves. It plays a key role in data science, from probability distributions to optimization.

What is an Integral?

An integral is the mathematical process of finding the total accumulation of a quantity. You can think of it as finding the area under a curve. If differentiation gives the rate of change, integration reconstructs the original function.

The basic integral of a power function follows this rule:

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:

Example:

Exponential Rule

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

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

Example:

Trigonometric Integrals

Sine and cosine functions also follow straightforward integration rules:

Definite Integrals

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

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

Example:

This means the area under the curve $y = 2x$ from $x=0$ to $x=2$ is $4$.