Integral Calculus
Related Pages
Calculus: Integration
Calculus: Derivatives
Calculus Lessons
In these lessons, we introduce a notation for antiderivatives called the Indefinite Integral. We also give a list of integration formulas that would be useful to know.
Indefinite Integrals
An indefinite integral (or antiderivative) of a function f(x) is a function F(x) whose derivative is f(x).
It is written as:
\( \int f(x) dx = F(x) + C \)
where
- \( \int \) is the integral symbol
- \( f(x) \) is the integrand
- \( dx \) indicates integration with respect to \( x \)
- \( F(x) \) is the antiderivative of \( f(x) \)
- C is the constant of integration
The notation \( \int f(x) dx \) is used for an antiderivative of f and is called the indefinite integral.
\( \int f(x) dx = F(x) + C \) means that \( F’(x) = f(x) \)
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
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:
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
Example:
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
Example:
∫3x5, dx
More Indefinite Integral, Step By Step, Examples: With Square Root
Example:
∫3√x, dx
More Indefinite Integral, Step By Step, Examples: x In The Denominator
Example:
∫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?
Example:
- ∫(3x2 - 2x + 1) dx
- ∫3ex dx
- ∫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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