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Math Help - integrating exponential

  1. #1
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    integrating exponential

    Um is this right:
    The integral of e^(1/4 x^2 s) dx
    = e^(s x^2 /4) x?

    Thank you
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  2. #2
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    Quote Originally Posted by gconfused View Post
    Um is this right:
    The integral of e^(1/4 x^2 s) dx
    = e^(s x^2 /4) x?

    Thank you
    I take it you mean \int e^{\frac{sx^2}{4}} dx.

    If s is a constant - ie. just a number - then the integral would be:

    \frac{2}{sx} e^{\frac{sx^2}{4}} + C
    Last edited by craig; May 26th 2009 at 05:49 AM. Reason: +C
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    Thanks!! Coz i tried writing it up in mathematica to check my answer, and it gave me that and that just looked dodgy. yeahh must have typed it wrong... thanks again
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  4. #4
    Super Member craig's Avatar
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    Just a little tip on integrating exponentials.

    If the question is \int e^{f(x)} dx

    Then the answer is \frac{1}{f'(x)} e^{f(x)} + C

    For example, \int e^{9x} dx would be \frac{1}{9} e^{9x} + C

    You can of course check this by differentiating your answer and see if you get your original equation.

    NOTE: This only works if the f(x) is a linear equation.
    Last edited by craig; May 27th 2009 at 05:21 AM. Reason: Thank you to mr fantastic
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  5. #5
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    Quote Originally Posted by craig View Post
    I take it you mean \int e^{\frac{sx^2}{4}} dx.

    If s is a constant - ie. just a number - then the integral would be:

    \frac{2}{sx} e^{\frac{sx^2}{4}} + C
    Sorry but if that's the question then this answer is totally wrong. Differentiate it and see for yourself.

    \int e^{\frac{sx^2}{4}} dx has no elementary primative.

    Quote Originally Posted by craig View Post
    Just a little tip on integrating exponentials.

    If the question is \int e^{f(x)} dx

    Then the answer is \frac{1}{f'(x)} e^{f(x)} + C

    For example, \int e^{9x} dx would be \frac{1}{9} e^{9x} + C

    You can of course check this by differentiating your answer and see if you get your original equation.
    Sorry but in general this is totally wrong. It is only correct when f(x) is a linear function.
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  6. #6
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    Quote Originally Posted by mr fantastic View Post
    Sorry but if that's the question then this answer is totally wrong. Differentiate it and see for yourself.

    \int e^{\frac{sx^2}{4}} dx has no elementary primative.


    Sorry but in general this is totally wrong. It is only correct when f(x) is a linear function.
    So what do you mean by no elementary primative? Because i did try and integrate it, and got the same answer as that guy. If that's not right, then how would i go about answering that question?
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  7. #7
    Super Member craig's Avatar
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    Quote Originally Posted by mr fantastic View Post
    Sorry but if that's the question then this answer is totally wrong. Differentiate it and see for yourself.

    \int e^{\frac{sx^2}{4}} dx has no elementary primative.


    Sorry but in general this is totally wrong. It is only correct when f(x) is a linear function.
    I was not aware of this , didn't know that the function of x had to be linear, I apologise.
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  8. #8
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    Quote Originally Posted by gconfused View Post
    So what do you mean by no elementary primative? Because i did try and integrate it, and got the same answer as that guy. If that's not right, then how would i go about answering that question?
    I mean that the answer to \int e^{\frac{sx^2}{4}} \, dx cannot be expressed as a finite number of elementary functions. If you post the details of your calculation I will point out your mistakes. Did you try differentiating your answer? Do you get e^{sx^2/4}?

    If this integral arises as a result of a question you're trying to answer, I suggest you post the entire question exactly as it's written.
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    Quote Originally Posted by gconfused View Post
    So what do you mean by no elementary primative? Because i did try and integrate it, and got the same answer as that guy. If that's not right, then how would i go about answering that question?
    It is not correct because the derivative of \frac{2}{sx}e^{sx^2/4} has to be done by the product rule: (fg)'= f'g+ fg'. The derivative of e^{sx^2/4} is \frac{2sx}{4}e^{sx^2/4} and that, multiplied by \frac{2}{sx} will give e^{sx^2/4} but then the derivative of \frac{2}{sx} is -\frac{2}{sx^2} so the derivative of the entire \frac{2}{sx}e^{sx^2/4} is e^{sx^2/4}- \frac{2}{sx^2}e^{sx^2/4}, not just e^{sx^2/4}.

    Craig's advice for integrating e^{f(x)} in general is not correct because what he is trying to do is use the substitution u= f(x). Then du= f'(x)dx so dx= du/f'(x). Putting those into the integral we have \int e^u\left(du/f'(x)\right) and, apparently, he is taking that f'(x) outside the integral. But if f is not linear, then f' is a function of x and cannot be taken outside the integral- it must be integrated also. If f were linear, that is, if f(x)= ax+ b then f'(x)= a, a constant and we can do that: Letting u= ax+ b, du= adx so dx= dx/a and \int e^{ax+ b}dx= \int e^u (du/a)= \frac{1}{a}\int e^u du = \frac{1}{a}e^u+ C= \frac{1}{a}e^{ax+b}+ C. But that is only possible with a being a constant.

    You should understand that most (in a very specific sense "almost all") integrable functions do NOT have "elementary functions" as integrals. (Elementary functions" are basically all the functions that you see in Calculus or lower level courses.)

    About the best you can do with \int e^{sx^2/4}dx is make the linear substitution u= \sqrt{s}x/2. Then du= \sqrt{s}/2 dx and dx= \frac{2}{\sqrt{s}} du so \int e^{sx^2/4} dx = \frac{2}{\sqrt{s}} \int e^{u^2} du = \frac{2}{\sqrt{s}} \text{erf}(x) where "erf(x)" is defined as \int e^{u^2}du. It is a non-elementary function, called the "error function" and is used extensively in statistics. In fact, I wouldn't be surprised if that problem came from a statistics course.
    Last edited by mr fantastic; May 27th 2009 at 06:08 AM. Reason: Fixed the last bit of latex
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