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5.2 The Natural Logarithmic Function: Integration

5.2 The Natural Logarithmic Function: Integration. After this lesson, you should be able to:. Use the Log Rule for Integration to integrate a rational function. Integrate trigonometric functions. Log Rule for Integration. In the last section, we have already learned that. 2. 1.

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5.2 The Natural Logarithmic Function: Integration

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  1. 5.2 The Natural Logarithmic Function: Integration

  2. After this lesson, you should be able to: • Use the Log Rule for Integration to integrate a rational function. • Integrate trigonometric functions.

  3. Log Rule for Integration In the last section, we have already learned that 2. 1. • Let u be a differentiable function of x. • Because du= u’dxthe 2nd formula can be written as It is easily concluded:

  4. Examples for Log Rule let u = 4x – 1, du = 4dx and x = (u +1)/4, then

  5. Examples of Quotient Form of the Log Rule

  6. Example 7 Find the area bounded by the following function, the x axis and the lines x= –1 and x=1: Note that Solution 0.424 sq units

  7. Example 8 Find the antiderivative of Note that Solution

  8. Example 9 Find the antiderivative of Let u = x + 1, then du = dx and x = u – 1 Solution

  9. Guidelines for Integration

  10. Summary: • Memorize the form • Make the problem fit the form • Integrate

  11. Example 10 Find the antiderivative of We notice that 1/xdx = d(lnx) Solution Apply the Log Rule for u = lnx

  12. Integrals of the Six Trig Functions

  13. Example 11 Find the antiderivative of We multiply and divide a factor (sec x + tan x) on the integrand, and we also notice that d(tan x) = sec2x dx and d(sec x) = sec x tan x dx Solution

  14. Example 11 Find the average value of on the interval of [0, π/4] Notice that sec x > 0 on the interval [0, π/4] Solution

  15. Homework Pg. 338 1-15 odd, 27-39 odd, 43-51 odd, 71

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