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Math Help - integral of constant raised to a power

  1. #1
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    integral of constant raised to a power

    Hi. What is the integral of a constant raised to a variable power? For example:

    \int{4^x dx}

    I imagined first that perhaps you could treat it like a variable raised to a constant power, but I'm not sure if that is correct, and my results didn't seem quite right when I looked at a graph of the function and an antiderivative together.
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  2. #2
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    Re: integral of constant raised to a power

    integral of constant raised to a power-formula.png
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  3. #3
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    Re: integral of constant raised to a power

    Quote Originally Posted by infraRed View Post
    Hi. What is the integral of a constant raised to a variable power? For example:

    \int{4^x dx}

    If y=4^x then y'=4^x(log(4)) so what is the answer to your question?
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    Re: integral of constant raised to a power

    It might be even easier to recognize that 4^x = e^{x\ln{4}}, which is e to a constant times x. I'm guessing you know how to integrate that....

    - Hollywood
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  5. #5
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    Re: integral of constant raised to a power

    I found in my calculus textbook a table of basic integration rules, which states that

    \int{a^u du} = (\frac{1}{\ln{a}})a^u + C

    Since I'm short on time, it's easiest just to go with that. I imagine that the hints provided above were leading me in that direction.
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  6. #6
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    Re: integral of constant raised to a power

    Yes,

    \int{4^x dx} = \int{e^{(\ln{4})x} dx} = \frac{1}{\ln{4}} e^{(\ln{4})x} + C = \frac{1}{\ln{4}} 4^x + C

    is correct.

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