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Math Help - stuck on integration by parts

  1. #1
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    stuck on integration by parts

    hello,

    im stuck on an integration by parts problem:

    \int{s \times 2^s}

    here is what i have so far

    u = s, du = ds
    dv = 2^{s}ds, v = \frac{2^{s+1}}{s+1}

    i know that

    \int{s \times 2^s} = uv - \int{vdu}
    = s \times \frac{2^{s+1}}{s+1} - \int{\frac{2^{s+1}}{s+1}du}

    but im not sure how to continue...

    could someone show me?
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  2. #2
    Rhymes with Orange Chris L T521's Avatar
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    Quote Originally Posted by absvalue View Post
    hello,

    im stuck on an integration by parts problem:

    \int{s \times 2^s}

    here is what i have so far

    u = s, du = ds
    dv = 2^{s}ds, v = \frac{2^{s+1}}{s+1}

    i know that

    \int{s \times 2^s} = uv - \int{vdu}
    = s \times \frac{2^{s+1}}{s+1} - \int{\frac{2^{s+1}}{s+1}du}

    but im not sure how to continue...

    could someone show me?
    Be careful!! \displaystyle\int 2^s\,ds\neq \frac{2^{s+1}}{s+1}!! You can't apply the power rule when your base is a number and your exponent is a variable!!

    However, observe that 2^s=e^{s\ln 2}.

    Can you proceed?
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  3. #3
    Junior Member
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    Ok, I think I understand...

    Is this correct then?

    \int \!s2^s\, ds

    Note that 2^s = e^{s\ln{2}}. Thus

    \int \!s2^s\, ds = \int \!se^{s\ln{2}}\,ds

    Let u=s and dv = e^{s\ln{2}}ds.

    Then du = ds and v = \frac{e^{s\ln{2}}}{\ln{2}}.

    Using integration by parts:

    \int \!se^{s\ln{2}}\,ds = uv - \int vdu

    = s\frac{e^{s\ln{2}}}{\ln{2}} - \int \!\frac{e^{s\ln{2}}}{\ln{2}}\,ds

    = s\frac{e^{s\ln{2}}}{\ln{2}} - \frac{e^{s\ln{2}}}{\ln^2{2}} + c

    = s\frac{2^s}{\ln{2}} - \frac{2^s}{\ln^2{2}} + c
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  4. #4
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    Yes, that is correct.

    It is also useful to learn as general rules: \frac{d a^x}{dx}= ln(a) a^x and \int a^x dx= \frac{a^x}{ln(a)}+ C.

    Notice that if a= e, ln(a)= ln(e)= 1 so that becomes the "usual" derivative and anti-derivative of e^x.
    Last edited by Chris L T521; June 28th 2010 at 07:56 PM. Reason: fixed LaTeX.
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