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Integral Calculus

In these lessons, we introduce a notation for antiderivatives called the Indefinite Integral. We will also give a list of integration formulas that would be useful to know.

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Indefinite Integrals

The notation is used for an antiderivative of f and is called the indefinite integral.

indefinite integral

The following is a table of formulas of the commonly used Indefinite Integrals. You can verify any of the formulas by differentiating the function on the right side and obtaining the integrand. Scroll down the page if you need more examples and step by step solutions of indefinite integrals.

Table of Indefinite Integral Formulas

Indefinite Integral Formulas


Find the general indefinite integral


Definite Integrals and Indefinite Integrals

The connection between the definite integral and indefinite integral is given by the second part of the Fundamental Theorem of Calculus.

If f is continuous on [a, b] then

definite integral

Take note that a definite integral is a number, whereas an indefinite integral is a function.




Definition of Indefinite Integrals
An indefinite integral is a function that takes the antiderivative of another function. It is visually represented as an integral symbol, a function, and then a dx at the end. The indefinite integral is an easier way to symbolize taking the antiderivative. The indefinite integral is related to the definite integral, but the two are not the same.

Antiderivatives and indefinite integrals
What is 2x the derivative of? This is the same as getting the antiderivative of 2x or the indefinite integral of 2x. Indefinite Integrals
Indefinite integrals are functions that do the opposite of what derivatives do. They represent taking the antiderivatives of functions.
A formula useful for solving indefinite integrals is that the integral of x to the nth power is one divided by n+1 times x to the n+1 power, all plus a constant term.

Indefinite integrals, step by step examples
Step 1: Add one to the exponent
Step 2: Divide by the same.
Step 3: Add C
1. ∫3x5, dx
More indefinite integral, step by step, examples: with square root
1. ∫3√x, dx
More indefinite integral, step by step, examples: x in the denominator
1. ∫6/x4, dx

Complicated Indefinite Integrals
Not all indefinite integrals follow one simple rule. Some are slightly more complicated, but they can be made easier by remembering the derivatives they came from. These complicated indefinite integrals include the integral of a constant (the constant times x), the integral of ex (ex) and the integral of x-1 (ln[x]).

Indefinite Integration (Polynomial, Exponential, Quotient)
How to determine antiderivatives using integration formulas?
1. ∫(3x2 - 2x + 1) dx
2. ∫3ex dx
3. ∫4/x dx Basic Integration Formulas
Here are some basic integration formulas you should know.

Definite Integral

The Definite Integral - Understanding the Definition. Calculating a Definite Integral Using Riemann Sums - Part 1.
This video shows how to set up a definite integral using Riemann Sums. The Riemann Sums will be computed in Part 2. Calculating a Definite Integral Using Riemann Sums - Part 2.


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