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Single variable Integration

  1. Jan 6, 2009 #1
    1. The problem statement, all variables and given/known data

    Hi could anyone please help me to solve the following integrals?
    x and z are the only variable in all of the following integrals. Others are just constants.

    (a) Integrate sin(2x) * [(sinx)^2] with limits [0,a] or with limits [-a, +a]

    (b) Integrate sin(3x) * [(sinx)^2] with limits [0,a]

    Does it require the same method for cos, for example cos(5x) * (cosx)^3

    (c) Integrate the indefinite integral 1/((x^2)+4)
    I guess this one has something to do with arctan, but I don't know how to start with it.
    Could anyone please show me the steps?

    Okay, I promise this is the last question....
    (d) Integrate the indefinite integral exp(-kz)/(z+iL) where k is a positive constant, such that z=iL is the singularity inside any simple closed contour C(R).

    I try to find the residue at z=-iL = (exp(-kz) / first derivative of x+iL and then evaluate everything at z=-iL which gives = exp(-ikL)

    Then the required integral is 2*pi*exp(ikL).
    But the question is, is that right or have I done anything fundamentally wrong?



    2. Relevant equations



    3. The attempt at a solution


    Thanks a lot, that's it from me!
     
  2. jcsd
  3. Jan 6, 2009 #2

    jgens

    User Avatar
    Gold Member

    Few brief comments.

    For a, I would recommend that you use the identity sin(2x) = 2sin(x)cos(x) and then make a proper choice for u

    For b, I believe a similar method using the identity sin(3x) = 3cos^2(x)sin(x) - sin^3(x) or a variant of it would work.

    For c, try the substitution 2u = x. Than x^2 = 4u^2. After that, integrate the resultant function.
     
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