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Properties of Integration

  1. Jan 13, 2014 #1

    Recently in class, my professor went over a relationship that exists between trigonometric functions, T(x), and their complementary functions. That is:

    ∫ T(x)dx = W(x) + C
    ∫ coT(x)dx = -co[W(x)] + C

    Without really providing a proof, we were told this relationship. I've plugged in numbers and tried graphing the scenario and it works from what I've done so far. If anyone could provide me a formal proof through any source, that would be great since I need a bit more convincing.

    Also, is a limitation to this property that both W(x) and T(x) must be a combination trigonometric functions themselves? Since when trying to solve ∫cscxdx using the answer from ∫secdx, an answer cannot be found. Are there any other limitations?

    Also, regarding the identities:
    sinAcosB = 1/2 [sin(A-B) + sin(A+B)
    sinAsinB = 1/2 [cos(A-B) - cos(A+B)]
    cosAcosB = 1/2[cos(A-B) + cos(A+B)]

    Is there a formal proof readily accessible for these identities?

  2. jcsd
  3. Jan 13, 2014 #2

    Simon Bridge

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    Comes from the definitions of the trigonometric functions on the unit circle... which tells you how they are related to each other.

    For the identities:
    Helps to noticing that for a point (x,y) in the complex plane, the vector from the origin to (x,y) makes an angle ##\theta## to the real axis:

    ##e^{i\theta}=\cos\theta+i\sin\theta = x+iy##

    You can use that for the integral of trig things too, since ##\int e^x dx = e^x +c##
  4. Jan 13, 2014 #3


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    Yes, all it takes is a little manipulation of the basic identities for the sine and cosine of the sum and difference of two angles:

  5. Jan 14, 2014 #4

    Simon Bridge

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