What Is an Antiderivative?
You've spent the first part of calculus learning how to find derivatives. Now we're going to reverse the process. An antiderivative of a function is a function whose derivative is the original function.
For example, the derivative of is . Therefore, is an antiderivative of .
But there is an important detail: an antiderivative is never just one function. Any constant can be added without changing the derivative, because the derivative of a constant is zero.
So if you're asked to find an antiderivative, your answer should almost always contain a constant of integration, written as .
The Connection Between Derivatives and Integrals
Differentiation and integration are essentially opposite processes. Differentiation takes a function and finds its rate of change. Integration, in its simplest form, works backward to recover a function from its derivative.
This is the basic idea behind the Fundamental Theorem of Calculus. For now, think of an indefinite integral as a way of asking: "What function would have this derivative?"
Indefinite Integrals
The notation for an antiderivative is an indefinite integral:
The symbol means "find an antiderivative of." The expression is called the integrand, and tells us that the variable of integration is .
For example:
Why? Because differentiating gives .
The Power Rule for Integration
The most important basic integration rule is the power rule for integration. It is closely related to the power rule for differentiation, but notice the crucial difference: when integrating a power, you increase the exponent by 1 and then divide by the new exponent.
This rule works as long as . The case is special because it produces a natural logarithm instead.
A Step-by-Step Method
- Rewrite the integrand as powers when helpful.
- Apply the appropriate integration rule. For powers, increase the exponent by 1 and divide by the new exponent.
- Add the constant of integration, .
- Differentiate your answer to check that you recover the original integrand.
Basic Antiderivative Rules
Constant Rule
A constant integrates to the constant multiplied by :
Constant Multiple Rule
A constant can be pulled outside an integral:
Sum and Difference Rules
You can integrate each term separately:
The Logarithm Rule
When the integrand contains , the power rule does not apply. Instead:
Basic Exponential and Trigonometric Rules
Several familiar derivatives give especially simple antiderivatives:
Worked Examples
Find .
Find .
Find .
Find .
Suppose and . Find .
Why the Constant of Integration Matters
Suppose you differentiate , , and . You get exactly the same result each time:
That means all of those functions are antiderivatives of the same function. The constant represents the entire family of possible antiderivatives.
Writing instead of gives only one member of the family of antiderivatives. Unless an initial condition determines the constant, include .
When differentiating, the exponent comes down and then decreases by 1. When integrating, the exponent increases by 1 and you divide by the new exponent. For example, integrates to , not .
The power rule does not work for . That case must be handled with the logarithm rule: .
Integration is easy to verify. Differentiate your answer. If you get the original integrand back, your antiderivative is correct.
Practice Problems
Find .
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Find .
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Find .
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Find .
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Given and , find .
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The Big Idea
Antiderivatives are the bridge from differentiation to integration. The central question is simple:
Once you can recognize the basic antiderivative rules, indefinite integration becomes a systematic process. The next step is to connect these antiderivatives to definite integrals, accumulated area, and the Fundamental Theorem of Calculus.
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