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Indefinite Integrals or Antiderivatives

You should distinguish carefully between definite and indefinite integrals. A definite integral is a number, whereas an indefinite integral is a function (or family of functions).

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Antiderivative An antiderivative of a function f is a function F such that Ex. An antiderivative of is since

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Indefinite Integral The expression:

read “the indefinite integral of f with respect to x,” means to find the set of all antiderivatives of f. x is called the variable of integration Integrand Integral sign

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Constant of Integration

Every antiderivative F of f must be of the form F(x) = G(x) + C, where C is a constant. Notice Represents every possible antiderivative of 6x.

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Power Rule for the Indefinite Integral, Part I

Ex.

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Power Rule for the Indefinite Integral, Part II

Indefinite Integral of ex and bx

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Sum and Difference Rules

Ex. Constant Multiple Rule Ex.

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Table of Indefinite Integrals

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Fundamental Theorem of Calculus (part 1)

If is continuous for , then

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Fundamental Theorem of Calculus (part 2)

1. Derivative of an integral. 2. Derivative matches upper limit of integration. 3. Lower limit of integration is a constant.

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Fundamental Theorem of Calculus (part 2)

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The Fundamental Theorem of Calculus

Ex.

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The Fundamental Theorem of Calculus

Ex.

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First Fundamental Theorem:

1. Derivative of an integral. 2. Derivative matches upper limit of integration. 3. Lower limit of integration is a constant.

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1. Derivative of an integral.

2. Derivative matches upper limit of integration. 3. Lower limit of integration is a constant.

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The upper limit of integration does not match the derivative, but we could use the chain rule.

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The lower limit of integration is not a constant, but the upper limit is.

We can change the sign of the integral and reverse the limits.

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Neither limit of integration is a constant.

We split the integral into two parts. It does not matter what constant we use! (Limits are reversed.) (Chain rule is used.)

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More Ex’s on the FTC

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