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Math Help - Integral of 5^x

  1. #1
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    Integral of 5^x

    As title states, I cannot recall how to integrate 5^x. I've tried looking in the book but can't seem to locate any example of it. I know it has to do with "e" and "ln" but can't seem to remember exactly.

    Steps would be great.
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  2. #2
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    Quote Originally Posted by Beeorz View Post
    As title states, I cannot recall how to integrate 5^x. I've tried looking in the book but can't seem to locate any example of it. I know it has to do with "e" and "ln" but can't seem to remember exactly.

    Steps would be great.
    Hint:
    \frac{d}{dx}5^x = ln(5) \cdot 5^x

    -Dan
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  3. #3
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    Quote Originally Posted by topsquark View Post
    Hint:
    \frac{d}{dx}5^x = ln(5) \cdot 5^x

    -Dan
    Haha well that makes it easy. Was really looking for some type of general rule so that I will be able to do just about any and all.

    So the rule for any variable raised to a constant is to multiple it by ln(constant)??
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    Quote Originally Posted by Beeorz View Post
    As title states, I cannot recall how to integrate 5^x. I've tried looking in the book but can't seem to locate any example of it. I know it has to do with "e" and "ln" but can't seem to remember exactly.

    Steps would be great.
    Let 5^x=y

    taking natural log, x \ln5 = \ln y

    so, \ln 5.dx = \frac{y}{1} dy

    <br />
\int 5^x ~dx = \frac{1}{\ln 5} \int \frac{y}{y}~dy

    = \frac{1}{\ln 5} \int ~dy

    = \frac{1}{\ln 5} y +c

    = \frac{1}{\ln 5}(5^x) +c

    = \frac{5^x}{\ln 5}+c

    in general, \boxed{\int a^x ~dx =\frac{a^x}{\ln a} +c}, \text {where a is any constant}
    Last edited by Shyam; September 10th 2008 at 06:02 PM.
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