1 / 15

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.

teva
Download Presentation

5.2 The Natural Logarithmic Function: Integration

An Image/Link below is provided (as is) to download presentation Download Policy: Content on the Website is provided to you AS IS for your information and personal use and may not be sold / licensed / shared on other websites without getting consent from its author. Content is provided to you AS IS for your information and personal use only. Download presentation by click this link. While downloading, if for some reason you are not able to download a presentation, the publisher may have deleted the file from their server. During download, if you can't get a presentation, the file might be deleted by the publisher.

E N D

Presentation Transcript

  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

More Related