Antiderivative
An antiderivative is the inverse of a derivative.
Contents
Rules
The basic rules/identities are:
Rule |
Formulation |
Defined for... |
logarithms |
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exponentiation |
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exponentiation with a multiple |
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exponentiation (generalized) |
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polynomials |
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n != -1 (see the logarithm antiderivative in this case) |
constants |
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constant multiple |
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Rule |
Formulation |
Defined for... |
sine |
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|
cosine |
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secant squared |
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U-Substitution
Given a compound function to integrate, let u be the inner function. The integral can be solved by converting into terms of u and du.
Consider the generalized exponentiation antiderivative rule. Given ∫ef(x)dx to integrate, let u = f(x). The first derivative of u is f ' , so du = f ' dx. By factoring this out of the outer function and substituting it for du, the integral is converted. See the final calculation above.
If solving a definite integral, note that the bounds must also be recalculated as a* = f(a) and b* = f(b).