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Thread: 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 $\displaystyle u=e^x$ ---> $\displaystyle du=e^xdx$
    $\displaystyle I=\int sec(u) du$

    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 $\displaystyle u=e^x$ and recall that $\displaystyle \int\sec{u}du=\ln|\sec{u}+\tan{u}|+C$.
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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
    $\displaystyle \int e^x \sec(e^x) \, dx$

    $\displaystyle u = e^x$ ... proceed using substitution.


    $\displaystyle y = y_0 e^{kt}$

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

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