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Math Help - Help with Integral Problems

  1. #1
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    Help with Integral Problems

    I have three in particular, two of which I can't seem to get anywhere and another which I think I may have domain issues.

    -Just for reference, I can only solve these using u-substitution or through anti-derivatives that we've memorized. (i.e. sinx'=cosx, x^n'=n-1x^n-1, etc.)

    1.) \int^\frac{\pi}{4}_\frac{-\pi}{4} \frac{t^4tan(t)}{2+cos(t)}\,dx

    I've tried u-substitution for this one but when finding du/dx nothing seems to replace every t variable to simplify into something that's easy to integrate. I've tried rewriting but rewriting it makes it even more complicated. (I've tried rewriting tan(t) into sin(t)/cos(t), I've tried to split up the problem, but no avail, nothing seems to work) A few tips on how to start would greatly help!!


    2.) \int^1_0 \frac{e^x}{1+e^{2x}}\,dx

    On this one, I've tried u-substitution with the denominator & numerator portion of the function:

    u=1+e^2x
    du/dx=e^2x * 2
    1/2du=e^2x dx

    or

    u=e^x
    du=e^x dx

    Neither one replaces every x variable in the original function so I'm stuck there.

    3.) \int^{10}_{1} \frac{x}{x^2-4}\,dx

    This last one I was able to integrate but once I start evaluating for the x intervals it asks for I get a domain issue. After integrating and using u-substitution I get

    F(x) = 1/2 ln (x^2 - 4) from x-values 10 and 1.

    When plugging in 10 I get a positive value which is solvable; however, when plugging in 1 I get a negative value which isn't defined at ln(x).


    Nevermind, solved this one



    -All the help and assistance will be greatly appreciated!!!
    Last edited by dagbayani481; December 5th 2011 at 08:36 PM.
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  2. #2
    MHF Contributor Amer's Avatar
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    Re: Help with Integral Problems

    second one

     \int _{0}^{1} \frac{ e^x}{1+ e^{2x}} \; dx

    let  u = e^x \Rightarrow du = e^x dx \Rightarrow \frac{du}{u} = dx

     \int \frac{u}{1+u^2} \left(\frac{du}{u}\right)

     \int \frac{u}{u(1+u^2)} \; du

     \int \frac{1}{1+u^2} \; du = \tan ^{-1} (u) + C
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  3. #3
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    Re: Help with Integral Problems

    The first one cannot be solved in terms of the standard functions. The third one is undefined.
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  4. #4
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    Re: Help with Integral Problems

    Quote Originally Posted by princeps View Post
    The first one cannot be solved in terms of the standard functions. The third one is undefined.
    Thank you, I kinda of find it odd that our would teacher would assign that problem considering we haven't covered the material necessary to solve it, but then again, we had a similar issue with a different problem that could not be solved with standard functions, and she let us omit it . I'll just bring it to her attention tomorrow in class.
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  5. #5
    A Plied Mathematician
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    Re: Help with Integral Problems

    Quote Originally Posted by dagbayani481 View Post
    I have three in particular, two of which I can't seem to get anywhere and another which I think I may have domain issues.

    -Just for reference, I can only solve these using u-substitution or through anti-derivatives that we've memorized. (i.e. sinx'=cosx, x^n'=n-1x^n-1, etc.)

    1.) \int^\frac{\pi}{4}_\frac{-\pi}{4} \frac{t^4tan(t)}{2+cos(t)}\,dx

    I've tried u-substitution for this one but when finding du/dx nothing seems to replace every t variable to simplify into something that's easy to integrate. I've tried rewriting but rewriting it makes it even more complicated. (I've tried rewriting tan(t) into sin(t)/cos(t), I've tried to split up the problem, but no avail, nothing seems to work) A few tips on how to start would greatly help!!
    The integrand is an odd function, and you are integrating over a symmetric interval. Can you infer anything from that?
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