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Math Help - Tips for surviving calc II?

  1. #1
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    Tips for surviving calc II?

    Hi all. I just started calc 2 at the beginning of the spring semester. I flew through calc I, and I think I understand all the topics we've learned so far in calc 2 (parts, trig identities/trig sub, and partial fractions). I have trouble when I get to a problem that doesn't seem to follow any of those methods though.

    For example, I look at

    Integral 1/(e^x*sqrt(1-e^(-2x)))

    with confusion. Are there any tips for approaching a problem such as this? I feel like a trig sub should work, but I can't seem to simplify it down right. Other problems with square and cube roots also give me trouble. In short, I can do a problem if it follows the guidelines for the methods I've learned so far, but I have a lot of trouble if I have to modify the problem in some way to make it look like a different problem that I can work with. I'm wondering if my algebra skills are behind or if I'm just looking at these problems the wrong way. Any information is appreciated. Thank you.
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  2. #2
    Super Member General's Avatar
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    \int \frac{dx}{e^x\sqrt{1-e^{-2x}}}=\int \frac{dx}{e^x\sqrt{1-\frac{1}{e^{2x}}}}

    =\int \frac{dx}{e^x \sqrt{\frac{e^{2x}-1}{e^{2x}}}}

    =\int \frac{dx}{e^x \,\ \frac{ \sqrt{e^{2x}-1} }{ \sqrt{e^{2x}} }<br />
}

    =\int \frac{\sqrt{e^{2x}}}{e^x \sqrt{e^{2x}-1} }dx

    =\int \frac{\sqrt{(e^x)^2}}{e^x \sqrt{e^{2x}-1} }dx

    =\int \frac{e^x}{e^x \sqrt{ (e^x)^2-1 } }dx

    By using the substitution u=e^x, we get:

    \int \frac{du}{u\sqrt{u^2-1}}

    which is a well-known integral.
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  3. #3
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    Sometimes you have to fiddle with the equations. In that particular one, you would have to substitute twice to solve it. Substitute once to get it into a nice form for another substitution to solve.

    A hint is that (e^x)^2 is the same as e^{2x}. So bring e^x to the top of the line, and sub in u=e^{-x}.

    Of course, that assumes that your equation is:
    \int{\frac{1}{e^{x}\sqrt{1-e^{-2x}}}dx}

    Hope this helps a bit.

    [edit]: Or General's way seems good
    Last edited by Diemo; February 14th 2010 at 10:14 AM.
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  4. #4
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    Thanks to both of you for solving that one. Diemo, fiddling with an equation is what I have trouble with. I can look at a problem and realize that it needs to be changed, but sometimes I'm just not sure how to change it.

    Another example (you don't need to solve it for me) is

    Integral (xsinx)/((cosx)^3)

    I realize I have to change it, and I do change it to xtanx(secx)^2, but the x out in front gives me trouble. My point is this. When you look at a problem and see that you have to fiddle with it, is there a list of generalized steps that you follow to try and make it look like a solvable equation? The previous integral I posted just needed some algebra steps to make it solvable, so I'm starting to think I'm behind on algebra.
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  5. #5
    Super Member General's Avatar
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    Quote Originally Posted by vReaction View Post
    Thanks to both of you for solving that one. Diemo, fiddling with an equation is what I have trouble with. I can look at a problem and realize that it needs to be changed, but sometimes I'm just not sure how to change it.

    Another example (you don't need to solve it for me) is

    Integral (xsinx)/((cosx)^3)

    I realize I have to change it, and I do change it to xtanx(secx)^2, but the x out in front gives me trouble. My point is this. When you look at a problem and see that you have to fiddle with it, is there a list of generalized steps that you follow to try and make it look like a solvable equation? The previous integral I posted just needed some algebra steps to make it solvable, so I'm starting to think I'm behind on algebra.
    Calculus - كتب Google
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  6. #6
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    vReaction

    As far as I know, there are no real steps to switching about your various values to get a function thatit is possible to integrate. You just get to know what to do by practice and constant practice, and it still doesn't always work out for you.

    Maybe one rule is that where you can split it so that you have a simple function and also have a function that you know is the derivitive of something, think about using integration by parts. For example, see something like xtan(x)sec^2(x), and you know that tan(x) sec^2(x) is the derivitive of something, think about using integration by parts. But for general rules, I think practice is the only one, you get to recoginse wwhich functions take which substitution.

    Hope that helps

    [edit] General, thanks, I have never seen this before.
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  7. #7
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    Ok I see. Thanks to both of you for the information.
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  8. #8
    Super Member General's Avatar
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    nvm man.
    But as you said, the important thing is the practice.
    All of the written steps can not let you "catch" the idea of \int x^x ( ln(x) + 1 ) \,\ dx , as an example.
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