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Math Help - Substition and Integration By Parts

  1. #1
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    Post Substition and Integration By Parts

    Evaluate \int2cos(ln t)dt by using a substitution prior to integration by parts.

    Thanks for any help!
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  2. #2
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    Try u = ln(t). Then t = exp(u).
    du = dt/t so dt = exp(u)du.

    Now you've got an integral with
    cos(u)exp(u)du.

    Remember cos(u) = {exp(iu)+exp(-iu)}/2 ?

    Now it's all exponentials.
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  3. #3
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    2* \int{cos(ln(t))dt}

     u=ln(t) and  du=\frac{1}{t}

    2* \int{cos(u)(\frac{1}{t}du)}

    Take your u equation and solve for t now.

     t={e}^{-u}

    2* \int{e^{-u}*cos(u)du}

     u=cos(u)
     du=-sin(u)
     dv={e}^{-u}
    v=-e^{-u}

    Solve and then back substitute

    uv- \int (vdu)
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  4. #4
    Member mathemagister's Avatar
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    Use the substitution u=ln(t) and then use the formula

    \int e^{\alpha u} \cos(\beta u) du = \frac{e^{\alpha u} (\alpha \cos(\beta u)+\beta \sin(\beta u))}{\alpha^2+\beta^2}+C

    which you can calculate yourself, instead of memorizing. (I personally prefer the former because you don't need it that much.)
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