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I'm having trouble finding the integral using u-substitution.

  1. Jun 8, 2011 #1
    1. The problem statement, all variables and given/known data

    I have to integrate using u-substitution (probably).

    Ex. 1 Integrate (sin^4x)/(cos^6x)dx

    2. Integrate (2x)/(sqrt(e^(2x^2)-1))dx

    3. Integrate (cos^-1x)/(sqrt(1-x^2))dx

    Thank you !

    2. Relevant equations

    I do not want the solutions. I just need to be pointed in the right direction (i.e. I need you to help me start off)

    **It should be noted that I am doing a calculus II course (Integral Calc, mostly) in university, so it's not very advanced integrals that I'm doing. Basically what I know is how to integrate using u-substitution, and I know the integrals for the inverse trig functions (which is supposed to be relevant to examples 2 & 3), and that's what information I have to work with.


    **It should also be noted that I may just not know how to rewrite the equations before I can integrate them. I have trouble 'seeing through' the equation and automatically knowing which way I'm going to solve it.

    3. The attempt at a solution

    Ex. 1 I tried rewriting the equation using trig identities, e.g. (1-cos(x))/(1-sin(x))^3. I found this got me nowhere.
    I also tried rewriting it is (sin^4)(x)/(cos^4)(x)*1/cos(x), and rewriting and rewriting until I ended up with a big mess, so that got me nowhere as well.

    2. Here's my dilemma:
    -if I substitute e^2x for u, I end up needing an e to the power in my numerator, so that doesn't work out.
    -if I instead substitute 2x^2 for u, I end up with the e to the power of u on the bottom and I don't have a formula for that.

    3. I have no ideas on this one.
     
  2. jcsd
  3. Jun 8, 2011 #2

    Mark44

    Staff: Mentor

    This is the right approach, but you have an error. Your integrand is equal to tan4(x)sec2(x). That should suggest a pretty obvious substitution.
    I don't have any ideas just yet, but I'll think about this one.
    If u = cos-1(x), what is du?

    BTW, welcome to Physics Forums!
     
  4. Jun 8, 2011 #3

    SteamKing

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    For example 2, rewrite the integrand to remove the sqrt in the denominator, that is, express 1/sqrt(e^(2*x^2)-1) using the appropriate exponent. After doing this, see if the factor 2x would be useful in integration by parts.
     
  5. Jun 8, 2011 #4

    SteamKing

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    Check that last suggestion.

    See if the factor 2x would be useful in a u-substitution integration.
     
  6. Jun 8, 2011 #5
    Thanks all, but I still cannot find the solution to the second example.

    I let u=2x, so du=2dx
    Since the x is is still in the numerator, I say that also, x=u/2
    So I fill this in and I get Integral of (u/2)(1/(sqrt((e^u)-1))du



    I have not yet learned to do integration by parts, by the way.
     
  7. Jun 8, 2011 #6

    vela

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    The argument of the exponential has an x2 in it, right? So try u=x2 to try simplify that a bit. That's where you find the factor of 2x comes in handy.

    Then you might try a substitution like v=eu and see where that gets you. A lot of this you figure out by trial and error. As you do more problems, you'll start to get a feel for what works and what doesn't.
     
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