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Is there an error in my textbook?

  1. Oct 8, 2015 #1
    1. The problem statement, all variables and given/known data
    1444318474802610179946.jpg

    2. Relevant equations

    Doesn't the integral of sec x tan x equal sec x?

    3. The attempt at a solution

    Screenshot_2015-10-08-17-31-27.png
     
  2. jcsd
  3. Oct 8, 2015 #2

    Ray Vickson

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    Science Advisor
    Homework Helper

    Your textbook is correct. The attachment of your work is too small and blurry for me to read, so I will not even try. I will look at it if you type it out.
     
  4. Oct 8, 2015 #3
    Great, thanks :-) Isn't the integral of sec x tan x equal to sec x?
     
  5. Oct 8, 2015 #4

    Mark44

    Staff: Mentor

    Yes, but I don't see how this fact applies to your question.
     
  6. Oct 8, 2015 #5

    Ray Vickson

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    Homework Helper

    Yes, as you can verify by taking the derivative. Likewise, you can check the book's answer by differentiation (which is something you should always do whenever you integrate).
     
  7. Oct 8, 2015 #6
    Well, it is correct, but the textbook uses the fact that [itex]\frac{d\tan x}{dx} = \sec^2 x[/itex]. If you use [itex] \frac{d(\sec x \tan x)}{dx} = \sec x [/itex], then the first part of integration will be
    [tex] \int \sec^2 x \tan^2 x dx = \int \sec x \tan^2 x d(\sec x \tan x), [/tex]
    which is not that easy to calculate. However, when you use [itex]\frac{d\tan x}{dx} = \sec^2 x[/itex], then
    [tex] \int \sec^2 x \tan^2 x dx = \int \tan^2 x d(\tan x) = \dfrac{1}{3} \tan^3 x + C_1. [/tex]
     
  8. Oct 8, 2015 #7
    Thank you! I just went over it again and realized I forgot about the chain rule.
     
  9. Oct 8, 2015 #8
    I just realized as much myself:-p I forgot about the chain rule, so I intgrated (sec x tan x)2 to be (sec3 x)/3.
     
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