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Find Indefinite Integral

  1. Nov 27, 2009 #1
    Using U-substitution find the indefinite integral of:

    [sin(2x)/cos^4(2x)] dx

    So I do know that it will have to come out to it being ln... heres what i did so far
    ok so i made u= cos^4(2x)
    du= -8cos^3(2x)*sin(2x)dx....(just took the derivative of u and simplified it)

    so made sin(2x)dx= du/(-8cos^3(2x))...so i can substitute it into my equation.

    so it came out to be:
    du/(u*-8cos^3(2x))...but in using u-substitution, i should not have an x variable...

    So do i have to minipulate u=cos^4(2x) to get x by its self?
    I get x= .5cos^-1(4sqrt(x))

    It just seems like its sooooo complicated.. dont know.
     
  2. jcsd
  3. Nov 27, 2009 #2

    Mark44

    Staff: Mentor

    No, it definitely won't come out being ln(something).
    That's because you're making it too complicated by using the wrong substitution. Instead, use u = cos(2x).
     
  4. Nov 27, 2009 #3
    ok thanks...
    So u=cos(2x)
    -du/2= sin(2x) dx...then i substitute:

    =-1/2[integral]du/u^4
    where i get =-1/2[integral] u^-4du
    =-1/2*-1/3u^-3+C
    =1/6u^-3+C

    Is that correct?...then i can just substitute what u equals into the equation (since they started in terms of x, ill leave it in terms of x)
     
  5. Nov 27, 2009 #4

    Mark44

    Staff: Mentor

    Right. And after you undo your substitution you can check your answer. Its derivative should be [sin(2x)/cos^4(2x)].
     
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