Warm-Up Rewrite using log properties before differentiation...

Solution Rewrite:

Solution …then differentiate And simplify:

Try This Find the derivative: Don’t forget the chain rule

5-2: Natural Logs and Integration Objectives: Integrate logarithmic functions Integrate trig functions using the log rule ©2003 Roy L. Gover (www.mrgover.com)

Definition Since The Log Rule for Integration

Definition The alternate form for the Log Rule for Integration: The chain rule version.

Example Find the antiderivative:

Example Find the antiderivative:

Important Idea u’ = ln|4x-1|+c u How the log rule works… Let u=4x-1, then du=4 u’ = ln|4x-1|+c u

Example Find the antiderivative:

Try This Find the antiderivative:

Try This Find the area bounded by the following function, the x axis and the lines x=-1 and x=1: Leave your answer accurate to 3 decimal places. .424 sq units

Example Hint: Use long division then integrate:

Warm-Up Hint: Since degree of numerator degree of denominator,divide then integrate.

Example Find the antiderivative: Let u=x+1, then du=1dx and x=u-1

Important Idea Read Guidelines for Integration on page 335 of your text Summary: memorize the form make the problem fit the form integrate

Example Let u =? 3 Choices: x ln x xln x Remember...

Example Evaluate:

Try This Evaluate without using your calculator: Confirm with your calculator

Review See p. 47 of your text

Example Hint: use identity then let u=cos x

Try This Evaluate: Leave answer as an exact value

Integrals of Trig Functions Important Idea Integrals of Trig Functions Memorize the integrals at the top of page 337 in your text

Example

Try This Find the indefinite integral:

Example Evaluate:

Try This Evaluate: Leave your answer as an exact value in radical form.

Try This The voltage,E, of a circuit is given by . Find the average voltage as t ranges from 0 to 0.5 seconds. 1.379 volts

Lesson Close Describe how to use the log rule for integration.

Assignment 1. 338/11-17 odd, 19-26 all & 31-37 odd