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Math Help - integrate this by parts and substitution

  1. #1
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    integrate this by parts and substitution

    imgur: the simple image sharer
    for the first one,
    i tried:

    let e^x = u
    du/dx = e^x
    but then i couldn't put that back in (du = e^x dx is of no use)

    then I tried let
    e^2x = u

    du = 2e^2x dx
    putting it back in i had

    1/2 Integral of [ ucos(u^1/2)du ]

    let f = u let dg/du = cos(u^1/2)
    df/du = 1 g = ?? Got ugly here i think.
    Any ideas?
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  2. #2
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    Re: integrate this by parts and substitution

    Hey 99.95.

    Use the hint that e^x*e^x = e^2x where du = e^xdx and u = e^x.
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  3. #3
    MHF Contributor MarkFL's Avatar
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    Re: integrate this by parts and substitution

    Your first substitution attempt would give you:

    u=e^x\,\therefore\,du=e^x\,dx=u\,dx\,\therefore\,d  x=\frac{du}{u} and your integral becomes:

    \int u\cos(u)\,du

    Now try integration by parts.
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  4. #4
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    Re: integrate this by parts and substitution

    Thanks guys, got it!

    Could I get some help for the second one? I'm quite bewildered
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  5. #5
    MHF Contributor MarkFL's Avatar
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    Re: integrate this by parts and substitution

    Let:

    u=\sqrt{x}\,\therefore\,du=\frac{1}{2\sqrt{x}}\,dx  \,\therefore\,dx=2u\,du and the integral becomes:

    2\int ue^u\,du

    Now use integration by parts.
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  6. #6
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    Re: integrate this by parts and substitution

    Try introducing a SQRT(x)*SQRT(x) term and just check that this is OK for the integral (it should be, but you'll need to check for yourself just in case). So instead of finding e^(SQRT(x))dx find SQRT(x)e^(SQRT(x))/SQRT(x)dx (Also this should work because you only have a single point where this breaks down which means it should be OK).
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