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Trig Substitution with Cosine

  1. Jan 30, 2016 #1
    I was wondering if you could do a trig substitution with cosine instead of sine. All the textbooks I have referred to use a sine substitution and leave no mention as to why cosine substitution was not used. It seemed that it should work just the same, until I tried it for the following Fint [sqrt(9-x^2)]/ [x^2]. I checked to see if my answer differed by only a constant but that was not the case. I have attached pictures of my work. Can anyone tell me why it does not work? 20160130_164904.jpg 20160130_164853.jpg
  2. jcsd
  3. Jan 30, 2016 #2


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    If you use sine instead, you will end up with ##\sin^{-1}## in place of ##\cos^{-1}##. But the two expressions are related by ##\cos^{-1}x = \pi/2-\sin^{-1}x##.
  4. Jan 30, 2016 #3
    I know but why are the answers different? Is one less valid than the other?

    Btw the textbook which uses sine has the same answer but with -sin^-1(x/3) instead of cos^-1(x/3) like I have it.
  5. Jan 30, 2016 #4


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    That's exactly the point I addressed in my previous post. Replace -sin^-1(x/3) with the equation I wrote before. You will indeed have an additional ##\pi/2## but it's a constant and hence can be absorbed into the integration constant ##C##.
  6. Jan 30, 2016 #5


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    There's really nothing magic about using sin or cos. It just depends on what is more convenient for each case. As for signs, using the relevant relations from trigonometry - like the one that blue_leaf77 mentions, you can substitute sin for cos and vice versa and find the appropriate sign.
  7. Jan 30, 2016 #6
    The function being integrated is

    f(x) = √(9 - x2) / x2

    . This is defined for 0 < |x| ≤ 3.

    When making a substitution we want to choose an interval where f(x) makes sense, and the easiest one is 0 < x ≤ 3.

    We also want to choose a substitution that takes the same values that f(x) does over the interval of definition, and that's between 0 and 3.

    Each of y = 3 sin(x) and y = 3 cos(x) satisfy this condition, so either one can be used for the substitution.

    Using 3 sin(x) to substitute might be a tiny bit easier than cosine because its derivative is 3 cos(x), and this does not introduce negative signs.
  8. Jan 30, 2016 #7
    Ok...I see it now. I tried some test bounds of integration and got the same answer from both options.
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