Basic Integration Formulas
Fundamental formulas for calculating integrals
Introduction
Basic integration formulas are fundamental to solving integral calculus problems. These formulas provide direct solutions for common integrals and serve as building blocks for solving more complex integrals. Mastering these formulas is essential for any calculus student or engineer.
On this page, we present the most important integration formulas, organized by categories for easy reference. Each formula includes a brief explanation and, in some cases, important conditions or restrictions.
Power Rule
General Formula
This formula applies to any power of x, except when n = -1 (which corresponds to the integral of 1/x).
Special Case: n = -1
This is the formula for the integral of 1/x. The absolute value is important to ensure the formula is valid for both positive and negative x.
Examples
Integral of x²
Integral of square root of x
Exponential and Logarithmic Functions
Exponential Function
The natural exponential function e^x is its own derivative, so its integral is also e^x.
General Exponential Function
For any positive base a (except a = 1), this formula provides the integral of a^x.
Logarithmic Function
This formula is derived using integration by parts and is fundamental for many applications.
Trigonometric Functions
Basic Trigonometric Functions
Powers of Trigonometric Functions
Inverse Trigonometric Functions
These formulas are particularly useful when working with trigonometric substitutions and appear in many physical and engineering applications.
Hyperbolic Functions
Hyperbolic functions appear in many physics and engineering problems, such as in the study of electromagnetic fields and fluid mechanics.
Special Forms
These special forms frequently appear in applications and are useful for solving integrals through substitutions.
Fundamental Formula
where
This is the fundamental formula of integral calculus, stating that the integral of a function is its antiderivative plus a constant.