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Warm-Up Rewrite using log properties before differentiation...
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Solution Rewrite:
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Solution …then differentiate And simplify:
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Try This Find the derivative: Don’t forget the chain rule
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5-2: Natural Logs and Integration
Objectives: Integrate logarithmic functions Integrate trig functions using the log rule ©2003 Roy L. Gover (
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Definition Since The Log Rule for Integration
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Definition The alternate form for the Log Rule for Integration:
The chain rule version.
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Example Find the antiderivative:
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Example Find the antiderivative:
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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
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Example Find the antiderivative:
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Try This Find the antiderivative:
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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
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Example Hint: Use long division then integrate:
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Warm-Up Hint: Since degree of numerator degree of denominator,divide then integrate.
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Example Find the antiderivative: Let u=x+1, then du=1dx and x=u-1
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Important Idea Read Guidelines for Integration on page 335 of your text Summary: memorize the form make the problem fit the form integrate
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Example Let u =? 3 Choices: x ln x xln x Remember...
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Example Evaluate:
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Try This Evaluate without using your calculator:
Confirm with your calculator
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Review See p. 47 of your text
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Example Hint: use identity then let u=cos x
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Try This Evaluate: Leave answer as an exact value
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Integrals of Trig Functions
Important Idea Integrals of Trig Functions Memorize the integrals at the top of page 337 in your text
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Example
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Try This Find the indefinite integral:
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Example Evaluate:
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Try This Evaluate: Leave your answer as an exact value in radical form.
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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
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Lesson Close Describe how to use the log rule for integration.
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Assignment 1. 338/11-17 odd, all & odd
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