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Math Help - Integral

  1. #1
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    Integral

    Hi,
    I am trying to solve ∫ln(t)^2 dt. I already solved part of the integral, however I don't know how to complete the exercise. Can someone help me? Thanks.

    ∫ln(t)^2 dt
    u=ln(t)^2 dv=dt
    du=2ln(t)*(1/t)dt v =t

    t*ln(t)^2-∫t*2ln(t)*(1/t)dt
    t*ln(t)^2-∫(t*2ln(t)/t) dt
    t*ln(t)^2-∫(2ln(t)) dt
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  2. #2
    MHF Contributor
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    Using the fact that \frac{d}{dt} t \ln t = 1 + \ln t,

    we have:

    \int 2 \ln t \cdot dt = 2 \int \ln t \cdot dt = 2 \left( \int (1 + \ln t) dt - \int dt \right) = 2t \ln t - 2 \int dt

    Does this help?
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  3. #3
    MHF Contributor chisigma's Avatar
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    Integrating by parts You obtain...

    \int \ln^{2} t \cdot dt = t\cdot \ln^{2} t - 2\cdot \int \ln t\cdot dt (1)

    ... and integrating by parts again You obtain...

    \int \ln t\cdot dt = t\cdot \ln t - \int dt= t\cdot \ln t - t + c (2)

    ... so that is...

    \int \ln^{2} t \cdot dt = t\cdot \ln^{2} t -2\cdot t\cdot \ln t +2\cdot t + c (3)

    Kind regards

    \chi \sigma
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  4. #4
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