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Math Help - Integration: General Power Formula

  1. #1
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    Integration: General Power Formula

    Hi. Question - integrate the following function:

    28-1-19

    (int) (4 + ((e^x))^3) e^x dx

    The answer is 1/4(4 + (e^x))^4 + C

    doing the work myself, du = 3((4 + (e^x))^2) which means it does not fit the general power formula, because e^x is not the derivative of (4 + ((e^x))^3)

    nor would it work the other way around, the derivative of e^x is e^x correct??

    or can someone explain how it does? Thanks.
    Last edited by togo; April 30th 2013 at 02:54 PM.
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  2. #2
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    Re: Integration: General Power Formula

    Hello, togo!

    You are choosing the wrong substitution.


    \int (4+e^x)^3\,e^x\,dx

    We have: . \int(4 + e^x)^3(e^x\,dx)

    Let u \,=\,4+e^x \quad\Rightarrow\quad du \,=\,e^x\,dx

    Substitute: . \int u^3\,du \;=\;\tfrac{1}{4}u^4+C

    Back-substitute: . \tfrac{1}{4}(4 + e^x)^4 + C

    Thanks from topsquark
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    Re: Integration: General Power Formula

    Quote Originally Posted by togo View Post
    Hi. Question - integrate the following function:

    28-1-19

    (int) (4 + ((e^x))^3) e^x dx

    The answer is 1/4(4 + (e^x))^4 + C

    doing the work myself, du = 3((4 + (e^x))^2) which means it does not fit the general power formula, because e^x is not the derivative of (4 + ((e^x))^3)

    nor would it work the other way around, the derivative of e^x is e^x correct??

    or can someone explain how it does? Thanks.
    If I am reading your integral right it is:
    \int \left (4 + \left ( e^x \right ) ^3 \right ) e^x dx

    Now \left ( e^x \right )^3 = e^{3x} so your integral becomes:
    \int \left (4 + \left ( e^x \right ) ^3 \right ) e^x dx = \int \left (4 + e^{3x} \right ) e^x dx

    Now try the substitution y = e^x.

    Just a thought: If the integral is supposed to be \int \left ( 4 + e^x \right )^3 e^x dx again do the substitution y = e^x.

    -Dan
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    Re: Integration: General Power Formula

    looks like I got the brackets wrong, thanks for extrapolating the answer. I see that I used chain rule instead of general power formula.
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    Re: Integration: General Power Formula

    Integration: General Power Formula-01-may-13-2.png
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