Indefinite Integrals or Antiderivatives

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6 Integration Antiderivatives and the Rules of Integration

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Presentation transcript:

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).

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

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

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.

Power Rule for the Indefinite Integral, Part I Ex.

Power Rule for the Indefinite Integral, Part II Indefinite Integral of ex and bx

Sum and Difference Rules Ex. Constant Multiple Rule Ex.

Table of Indefinite Integrals

Fundamental Theorem of Calculus (part 1) If is continuous for , then

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.

Fundamental Theorem of Calculus (part 2)

The Fundamental Theorem of Calculus Ex.

The Fundamental Theorem of Calculus Ex.

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

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

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

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.

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.)

More Ex’s on the FTC