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Math Help - Need Help Starting Problem

  1. #1
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    Need Help Starting Problem

    Evaluate the integral by making a substitution that converts the integrand to a rational function
    \int\ \frac{e^t}{e^{2t}-4}dx
    can i do the problem this way
    \int\ \frac{e^t}{e^{t^2}-4}dx
    then doin a u sub with u=e^t
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  2. #2
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    Quote Originally Posted by vinson24 View Post
    Evaluate the integral by making a substitution that converts the integrand to a rational function
    \int\ \frac{e^t}{e^{2t}-4}dx
    can i do the problem this way
    \int\ \frac{e^t}{e^{t^2}-4}dx
    then doin a u sub with u=e^t
    \int {\frac{{du}}<br />
{{u^2  - 4}} = \frac{1}<br />
{4}\int {\left( {\frac{1}<br />
{{u - 2}} - \frac{1}<br />
{{u + 2}}} \right)} du}
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  3. #3
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    Quote Originally Posted by Plato View Post
    \int {\frac{{du}}<br />
{{u^2 - 4}} = \frac{1}<br />
{4}\int {\left( {\frac{1}<br />
{{u - 2}} - \frac{1}<br />
{{u + 2}}} \right)} du}
    How did you get the 1/4?
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  4. #4
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    \frac{1}<br />
{{u - 2}} - \frac{1}<br />
{{u + 2}} = \frac{4}<br />
{{u^2  - 4}}
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  5. #5
    MHF Contributor harish21's Avatar
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    Quote Originally Posted by vinson24 View Post
    Evaluate the integral by making a substitution that converts the integrand to a rational function
    \int\ \frac{e^t}{e^{2t}-4}dx
    can i do the problem this way
    \int\ \frac{e^t}{e^{t^2}-4}dx
    then doin a u sub with u=e^t
     \int {\frac{e^y}{-4+e^{2t}}} dt

    substitute  u = e^t and du = e^{t} dt :

    = \int {\frac{1}{u^{2}-4}} du


    =  \frac{-1}{2} tanh^{-1}\frac{u}{2}+C

    Substitute back for u = e^t:

    = \frac{-1}{2}tanh^{-1}\frac{e^t}{2}+C

    Which is equivalent for t values to:

    = \frac{1}{4} log(e^{t}-2)-\frac{1}{4} log(e^{t}+2)+C
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