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Aim: How do we integrate the natural logarithmic function?

Aim: How do we integrate the natural logarithmic function?. Do Now:. Log Rule for Integration. Rules of Differentiation. Rules of Integration. Enables integration of rational functions. Model Problems. u = 4 x – 1. u’ = 4. Multiple & Divide by 4. Substitute u. Log rule.

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Aim: How do we integrate the natural logarithmic function?

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  1. Aim: How do we integrate the natural logarithmic function? Do Now:

  2. Log Rule for Integration Rules of Differentiation Rules of Integration Enables integration of rational functions

  3. Model Problems u = 4x – 1 u’ = 4 Multiple & Divide by 4 Substitute u Log rule Back substitute

  4. Model Problem Alternate form of Log Rule du = u’dx Look for quotients in which numerator is the derivative of denominator.

  5. Look for quotients in which numerator is derivative of denominator. Model Problem u = x2+ 1 u’ = 2x

  6. Look for quotients in which numerator is derivative of denominator. u = x3+ x u = tanx u = x2 + 2x u’ = 3x2 + 1 u’ = sec2x u’ = 2x+ 2 Model Problems

  7. Look for quotients in which numerator is a degree higher or equal to denominator u = x2 + 1 u’ = 2x Model Problems long division

  8. Aim: How do we integrate the natural logarithmic function? Do Now:

  9. Look for quotients in which numerator is derivative of denominator. Model Problem – Change of Variables u = x + 1 x = u – 1 Substitute u Rewrite 2 fractions Rewrite 2 Integrals

  10. Look for quotients in which numerator is derivative of denominator. Model Problem – Change of Variables Rewrite 2 Integrals Integrate Simplify Back- substitute

  11. Guidelines for Integration • 1. Memorize a basic list of integration formulas. (20) • Find an integration formula that resembles all or part of the integrand, and, by trial and error, find a choice of u that will make the integrand conform to the formula. • If you cannot find a u-substitution that works, try altering the integrand. You might try a trig identity, multiplication and division by the same quantity, or addition and subtraction of the same quantity. Be creative. • If you have access to computer software that will find antiderivatives symbolically, use it.

  12. Model Problem Will log rule apply? What does u equal? u = x u = lnx u = x lnx u’ = 1/x Divide N & D by x Substitute u Log Rule Back- substitute

  13. Integrals of Trig Functions u = cosx u’ = -sinx Substitute u Log Rule Back- substitute

  14. Integrals of Trig Functions u = sec x + tan x u’ = sec x tanx + sec2x Log Rule Back- substitute

  15. Integrals for Basic Trig Functions

  16. Model Problem 1 + tan2x = sec2x - Pythagorean Identity

  17. Model Problem The electromotive force E of a particular electrical circuit is given by E = 3sin2t, where E is measured in volts and t is measured in seconds. Find the average value of E as t ranges from 0 to 0.5 second. u = 2t du = 2dt  1.379 volts

  18. Aim: How do we integrate the natural logarithmic function? Do Now:

  19. Model Problem

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