Introduction to Integration

What is Integration?

Integration is a fundamental concept in calculus, often described as the reverse process of differentiation. It is used to find areas under curves, among other applications.

In simple terms, if differentiation gives us the rate of change, integration helps us find the total change or accumulation.

For example, if you know the speed of a car at different times, integration can help you find the total distance traveled.

Basic Integration Notation

• The integral of a function $f(x)$ with respect to $x$ is written as: $$\int f(x) \, dx$$
• The symbol $\int$ is called the integral sign.
• $f(x)$ is the integrand.
• $dx$ indicates the variable of integration.

Indefinite Integrals

An indefinite integral is a family of functions and includes a constant of integration $C$. It is written as:

• $$\int f(x) \, dx = F(x) + C$$
• Where $F(x)$ is the antiderivative of $f(x)$.

Basic Rules of Integration

• Constant Rule: $\int a \, dx = ax + C$ where $a$ is a constant.
• Power Rule: $\int x^n \, dx = \frac{x^{n+1}}{n+1} + C$ for $n \neq -1$.
• Sum Rule: $\int [f(x) + g(x)] \, dx = \int f(x) \, dx + \int g(x) \, dx$.