5 2 Antiderivatives Indefinite Integrals Objectives Define the

5 -2: Antiderivatives & Indefinite Integrals Objectives: • Define the antiderivative (indefinite integral) • Learn basic antidifferentiation rules © 2002 Roy L. Gover (roygover@att. net)

Definition If F(x) is an antiderivative of f(x) on an interval I, then F’ (x)=f(x) + c where c is a constant.

Example Let be an antiderivative of f(x) then

The Problem. . . Find the function given its derivative. The function is the antiderivative or indefinite integral.

Example If , find f(x), the function whose derivative is. Confirm your answer.

Try This If , find f(x), the function whose derivative is. Confirm your answer.

Try This If , find f(x), the function whose derivative is. Confirm your answer.

Try This If , find f(x), the function whose derivative is. Confirm your answer.

Try This If , find f(x), the function whose derivative is. aka the power rule for integration.

Example Find the antiderivative of 2 x 2.

Example Find the antiderivative of cos x.

Try This Find the antiderivative of sin x. -cos x

Definition Variable of Integration Integral Sign Integrand Constant of Integration

Example Find the antiderivative:

Definition Power Rule for Integration Special Case:

Definition Sum& Difference Rule for Integration The integral of the difference sum is is the difference of the sum of the integrals… integrals

Example Evaluate the indefinite integral:

Try This Find the antiderivative: Hint: re-write and use the power rule.

Try This Find the antiderivative: Hint: Re-write as 2 fractions.

Solution

Example Evaluate the indefinite integral:

Example Sometimes we want a particular antiderivative. . . Find the equation for y given that y’=2 x+3 and y passes through the point (2, 1)

Lesson Close See page 327 of your text for basic integration rules. e iz Since integration is the r o m inverse of differentiation, you e M most of the already know rules.