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Math Help - Integration and Carbon-14 help

  1. #1
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    Integration and Carbon-14 help

    Questions:
    Integrate e^(x)sec(e^(x))dx

    Carbon 14 question: How old is a painting when it has 99.5% of it's original carbon 14? (assuming 5700 is the half-life of carbon 14)

    I think you use u-substitution for the 1st one, but I'm not so sure what to do with that, so walk me through it. Also the answer for that is ln lsec(e^(x)+tan(e^(x))l+c (read as logarithm of the absolute value...etc)

    For the other one, am I supposed to memorize the constant for carbon-14? Thank you
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  2. #2
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    Quote Originally Posted by maximade View Post
    Questions:
    Integrate e^(x)sec(e^(x))dx
    Substitute u=e^x ---> du=e^xdx
    I=\int sec(u) du

    What is the value of the last integral?
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  3. #3
    No one in Particular VonNemo19's Avatar
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    Quote Originally Posted by maximade View Post
    Questions:
    Integrate e^(x)sec(e^(x))dx

    Carbon 14 question: How old is a painting when it has 99.5% of it's original carbon 14? (assuming 5700 is the half-life of carbon 14)

    I think you use u-substitution for the 1st one, but I'm not so sure what to do with that, so walk me through it. Also the answer for that is ln lsec(e^(x)+tan(e^(x))l+c (read as logarithm of the absolute value...etc)

    For the other one, am I supposed to memorize the constant for carbon-14? Thank you
    For the first. Let u=e^x and recall that \int\sec{u}du=\ln|\sec{u}+\tan{u}|+C.
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  4. #4
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    Quote Originally Posted by maximade View Post
    Questions:
    Integrate e^(x)sec(e^(x))dx

    Carbon 14 question: How old is a painting when it has 99.5% of it's original carbon 14? (assuming 5700 is the half-life of carbon 14)

    I think you use u-substitution for the 1st one, but I'm not so sure what to do with that, so walk me through it. Also the answer for that is ln lsec(e^(x)+tan(e^(x))l+c (read as logarithm of the absolute value...etc)

    For the other one, am I supposed to memorize the constant for carbon-14? Thank you
    \int e^x \sec(e^x) \, dx

    u = e^x ... proceed using substitution.


    y = y_0 e^{kt}

    \frac{1}{2} = e^{k \cdot 5700}

    solve for k, then find the value of t when y = .995 , y_0 will equal 1, of course.
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