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Simple, yet ambiguous integral question

  1. Mar 19, 2016 #1
    1. The problem statement, all variables and given/known data

    Evaluate the definite (from -0.3 to 0.3) integral ∫tanxdx

    2. Relevant equations

    (dy/dx)tanx=(secx)^2

    3. The attempt at a solution

    Using the anti-derivative, I got to (sec(0.3))^2-(sec(-0.3))^2, which gives approximately 2. However, if the derivative is sketched out, and the area underneath is found, the answer is then 0.

    I found this happening with other similar questions as well. What could be going on?
     
  2. jcsd
  3. Mar 19, 2016 #2

    ShayanJ

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    Gold Member

    ## \frac{d}{dx} \tan x=\sec^2 x ## implies ## \int \sec^2 x \ dx=\tan x##, not ## \int \tan x \ dx=\sec^2 x##!
    ##\tan x## is an odd function of x so its definite integral over -b to b, for any real b, is 0.
    Of course, for some b, the interval may contain a singular point of ##\tan x## in which case the definite integral would diverge.
     
  4. Mar 19, 2016 #3

    Ray Vickson

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    Science Advisor
    Homework Helper

    You have the wrong antiderivative, but that is not your only error: you should get an answer of 0 (not ##\approx## 2) because ##\sec^2(0.3) = \sec^2(-0.3)##.
     
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