Integral Calculus

In this lesson, 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.

 

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.

Table of Indefinite Integrals Formulas

Example:

Find the general indefinite integral

Solution:

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

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

 

Example:

Evaluate

Solution:



Videos

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

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.







More indefinite integral, step by step, examples: with square root



More indefinite integral, step by step, examples: x in the denominator







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]).

Basic Indefinite Integration (Polynomial, Exponential, Quotient)
This video provides three basic examples of how to determine antiderivatives using integration formulas.



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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