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Math Help - Trig substitution.. Please HELP !

  1. #1
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    Trig substitution.. Please HELP !

    I really need some help with these trig substitution problems. Please write the u and du values for each. I'd really really appreciate it !

    1) Integrate [e^x/(e^2x +1)] dx

    2) Integrate [X^5 (lnx)] dx

    3) Integrate [sinx*cosx] dx

    Thank you Again !!
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  2. #2
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    Quote Originally Posted by qzack96 View Post

    3) Integrate [sinx*cosx] dx
    \int \sin(x)\cos(x) dx

    u = \sin(x) \Rightarrow \frac{du}{dx} = \cos(x)

    \int u \frac{du}{dx}dx

    \int u~du

    can you take it from here?
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  3. #3
    MHF Contributor matheagle's Avatar
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    1 let  u=e^x, so your integral is now

    \int {du\over u^2+1}=\arctan u+C=\arctan (e^x)+C

    2 Parts with  u=\ln x and dv=x^5

    3 can be done several ways,  u=\sin x, w=\cos x or use \sin (2x)=2\sin x\cos x

    The last way makes this

     {1\over 2}\int \sin (2x)dx={-1\over 4} \cos (2x)+C
    Last edited by matheagle; July 29th 2009 at 10:18 PM.
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  4. #4
    Junior Member enjam's Avatar
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    For 1, we can divide both sides by e^x to get:
    ∫ 1 / (e^x + e^-x) . dX
    Have you been taught that cosh(x) = 0.5 (e^x + e^-x)?
    So we have:
    ∫ 1 / 2cosh(x) . dx
    = 0.5 ∫ 1 / cosh(x) . dx
    = 0.5 ∫ sech(x) . dx
    Are you able to go on from here or have I just made it even more complicated?
    Last edited by enjam; July 30th 2009 at 02:30 AM. Reason: typo
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  5. #5
    Senior Member DeMath's Avatar
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    Quote Originally Posted by qzack96 View Post
    I really need some help with these trig substitution problems. Please write the u and du values for each. I'd really really appreciate it !

    2) Integrate [X^5 (lnx)] dx

    Thank you Again !!
    \int {{x^5}\ln xdx}  = \frac{1}{6}\int {\ln xd\left( {{x^6}} \right)}  = \frac{{{x^6}}}{6}\ln x - \frac{1}{6}\int {{x^6}d\left( {\ln x} \right)}  =

     = \frac{{{x^6}}}{6}\ln x - \frac{1}{6}\int {\frac{{{x^6}}}{x}dx}  = \frac{{{x^6}}}{6}\ln \left| x \right| - \frac{1}{6}\int {{x^5}dx}  = \frac{{{x^6}}}{6}\ln \left| x \right| - \frac{{{x^6}}}{{36}} + C.
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