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Math Help - Integration by parts problem

  1. #1
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    Integration by parts problem

    \int (e^(2x))(tan(e^(2x)))^2 dx

    I've been trying to integrate this by parts, and I'm failing miserably because no matter how I do it, I just make the integral more complicated or get something that doesn't make any sense at all, so it's pretty useless to put up any attempts at this problem.
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  2. #2
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    Use substitution rather than parts.
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    Do you HAVE to use "parts"? That just wouldn't be my first guess. Did you try u = e^(2x)?

    One may wish to recall that [tan(x)]^2 + 1 = [sec(x)]^2 and d(tan(x))/dt = [sec(x)]^2
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    Alright, I'm pretty rusty on the details of substitution since we learned that several months ago lol but I've got this so far

    \int (e^(2x))(tan(e^(2x)))^2 dx
    \int (tan u)^2 du

    and now im stuck again. If I split tan^2 u into sin^2 u/cos^2 u, then I can't figure out what to substitute next. Sorry I need so much help on this problem, I'm pretty embarrassed considering that (tan u)^2 isn't that elaborate of a function :/

    Thanks so much you guys!
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  5. #5
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    The integral of tan^2(x) is tan(x)-x+C, this is a common result.
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    Is there a way to arrive at that answer using a method of integration, or is that just something I'll have to memorize before the ap exam?
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  7. #7
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    Quote Originally Posted by mrantoine View Post
    Is there a way to arrive at that answer using a method of integration, or is that just something I'll have to memorize before the ap exam?
    Note that tan^2(x)= sec^2(x)-1

    It should hopefully be a little clearer now?!
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  8. #8
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    ohhh gotcha lol sorry, I'm so tired, I've been studying for 5 hours straight so far today.

    So the answer to the whole problem should be tan(e^(2x)) - e^(2x) + c, correct?

    p.s. you guys are awesome
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  9. #9
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    Quote Originally Posted by mrantoine View Post

    So the answer to the whole problem should be tan(e^(2x)) - e^(2x) + c, correct?
    This looks good, you can check it yourself by differentiating this.
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  10. #10
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    Quote Originally Posted by mrantoine View Post
    Alright, I'm pretty rusty on the details of substitution since we learned that several months ago lol but I've got this so far

    \int (e^(2x))(tan(e^(2x)))^2 dx
    \int (tan u)^2 du

    and now im stuck again. If I split tan^2 u into sin^2 u/cos^2 u, then I can't figure out what to substitute next. Sorry I need so much help on this problem, I'm pretty embarrassed considering that (tan u)^2 isn't that elaborate of a function :/

    Thanks so much you guys!
    Substitution isn't quite there. You appear to have picked "u" okay. What did you get for "du"? Something fishy going on in there.
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