Unit 1: Introduction to Integrals

In integral calculus, you’ll encounter two primary types of integrals:

indefinite integrals, [latex]\int f(x) \, dx[/latex],

and definite integrals, [latex]\int_{b}^{a} f(x) \, dx[/latex].

Indefinite integrals represent a family of functions rather than a specific value. They are expressed using the integral symbol without upper and lower limits. Indefinite integrals essentially answer the question “What function has a derivative equal to the integrand?”

On the other hand, definite integrals have specific bounds, indicating the range over which the integration is performed. They yield a single numerical value, representing the accumulated area under the curve between the specified limits.

The main difference lies in their outcomes: indefinite integrals produce functions, while definite integrals yield numerical values. For example

[latex]\int 3x^2 \, dx = x^3+C, \,\,\,\,\,\, \int_{1}^2 3x^2 \, dx = 7. [/latex]

We will learn the techniques how to solve indefinite and definite integrals.