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General rule for integrating e? (C3 Maths)

Hi

I know how to differentiate values of e with the general idea being:

Unparseable latex formula:

\frac{d}{dx} e^f ^(^x^)

becomes
Unparseable latex formula:

f'(x)e^f ^(^x^)



so what about:

Unparseable latex formula:

\int e^f^(^x^)

becomes ??? ???
(edited 13 years ago)
Reply 1
Avatar for nuodai
nuodai
17
There isn't a general rule for that. In fact, in the majority of cases, the integral of ef(x)e^{f(x)} isn't expressible in terms of elementary functions.

In general, ∫f′(x)ef(x) dx=ef(x)+C\displaystyle \int f'(x)e^{f(x)}\, dx = e^{f(x)}+C, which is a result derived from integration by substitution (which you might not see until C4).

However, in the case when f(x)=ax+bf(x) = ax+b (i.e. when it's a linear polynomial), we have the result ∫eax+b dx=eax+ba+C\displaystyle \int e^{ax+b}\, dx = \dfrac{e^{ax+b}}{a}+C, but we can't generalise this result to other functions.
Original post by nuodai

Original post by nuodai
There isn't a general rule for that. In fact, in the majority of cases, the integral of ef(x)e^{f(x)} isn't expressible in terms of elementary functions.

In general, ∫f′(x)ef(x) dx=ef(x)+C\displaystyle \int f'(x)e^{f(x)}\, dx = e^{f(x)}+C, which is a result derived from integration by substitution (which you might not see until C4).

However, in the case when f(x)=ax+bf(x) = ax+b (i.e. when it's a linear polynomial), we have the result ∫eax+b dx=eax+ba+C\displaystyle \int e^{ax+b}\, dx = \dfrac{e^{ax+b}}{a}+C, but we can't generalise this result to other functions.


And if it's not a linear polynomial?
Reply 3
Avatar for nuodai
nuodai
17
Original post by im so academic
And if it's not a linear polynomial?


Then it probably can't be expressed in terms of elementary functions. For example, you can't express ∫e−x2 dx\displaystyle \int e^{-x^2}\, dx in terms of elementary functions... nor can you express ∫esin⁡x dx\displaystyle \int e^{\sin x}\, dx or ∫eex dx\displaystyle \int e^{e^x}\, dx or ∫e1x dx\displaystyle \int e^{\frac{1}{x}}\, dx (etc...) in terms of elementary functions.
Original post by nuodai

Original post by nuodai
Then it probably can't be expressed in terms of elementary functions. For example, you can't express ∫e−x2 dx\displaystyle \int e^{-x^2}\, dx in terms of elementary functions... nor can you express ∫esin⁡x dx\displaystyle \int e^{\sin x}\, dx or ∫eex dx\displaystyle \int e^{e^x}\, dx or ∫e1x dx\displaystyle \int e^{\frac{1}{x}}\, dx (etc...) in terms of elementary functions.


What is an elementary function specifically? Wikipedia says it is "a function built from a finite number of exponentials, logarithms, constants, one variable, and nth roots through composition and combinations using the four elementary operations"?
Reply 5
Avatar for nuodai
nuodai
17
Original post by im so academic
What is an elementary function specifically? Wikipedia says it is "a function built from a finite number of exponentials, logarithms, constants, one variable, and nth roots through composition and combinations using the four elementary operations"?


Yup, that. This then includes, say, trig functions, since cos⁡x=eix+e−ix2\cos x = \dfrac{e^{ix} + e^{-ix}}{2}, and the modulus function, since ∣x∣=x2|x|=\sqrt{x^2}, and so on; but it doesn't include the integrals I mentioned above (despite them being perfectly nice functions).
Original post by nuodai

Original post by nuodai
Yup, that. This then includes, say, trig functions, since cos⁡x=eix+e−ix2\cos x = \dfrac{e^{ix} + e^{-ix}}{2}, and the modulus function, since ∣x∣=x2|x|=\sqrt{x^2}, and so on; but it doesn't include the integrals I mentioned above (despite them being perfectly nice functions).


So an elementary function is basically any function that's not an integral or a differential equation?
Reply 7
Avatar for vijaygupta2
vijaygupta2
exactly the same, except instead of multiplying by the differential [f'(x)], you DIVIDE by it.
Original post by vijaygupta2
exactly the same, except instead of multiplying by the differential [f'(x)], you DIVIDE by it.


So go integrate e^(-x^3) for me.
Reply 9
Avatar for nuodai
nuodai
17
Original post by vijaygupta2
exactly the same, except instead of multiplying by the differential [f'(x)], you DIVIDE by it.


Incorrect. If that were true, then when you differentiate ex22x\dfrac{e^{x^2}}{2x} you'd get ex2e^{x^2}, which you don't.
Reply 10
Avatar for nuodai
nuodai
17
Original post by im so academic
So an elementary function is basically any function that's not an integral or a differential equation?


No. Differential equations are equations, not functions, and some integrals can be expressed as elementary functions (e.g. the integral of sin⁡x\sin x)... to be honest, the whole notion of an elementary function isn't something you need to worry about at A-level (and aren't you doing GCSEs? In which case you really won't need to worry about them for a long time).
Reply 11
Avatar for purplefrog
purplefrog
OP
18
Original post by nuodai
There isn't a general rule for that. In fact, in the majority of cases, the integral of ef(x)e^{f(x)} isn't expressible in terms of elementary functions.

In general, ∫f′(x)ef(x) dx=ef(x)+C\displaystyle \int f'(x)e^{f(x)}\, dx = e^{f(x)}+C, which is a result derived from integration by substitution (which you might not see until C4).

However, in the case when f(x)=ax+bf(x) = ax+b (i.e. when it's a linear polynomial), we have the result ∫eax+b dx=eax+ba+C\displaystyle \int e^{ax+b}\, dx = \dfrac{e^{ax+b}}{a}+C, but we can't generalise this result to other functions.


thanks :smile:
Reply 12
Avatar for Calrik
Calrik
e^-x^2 or e^f(x) is one of those functions that can be solved by using polar spherical co-ordinates. But that's probably by the by it is a Gaussian though.

You can essentially "cheat" somewhat to find a general answer:

∫−∞∞e−x2 dx=π\int_{-\infty}^\infty e^{-x^2}\,dx=\sqrt{\pi}

∫−∞∞ae−(x−b)22c2 dx=ac⋅2π.\int_{-\infty}^\infty a e^{- { (x-b)^2 \over 2 c^2 } }\,dx=ac\cdot\sqrt{2\pi}.

It's a physics thing probably useless but just an aside. It's called an error function and it is only generally solvable in the case of an exception bounded by limits.
(edited 13 years ago)
Reply 13
Avatar for Calrik
Calrik
Original post by nuodai
No. Differential equations are equations, not functions, and some integrals can be expressed as elementary functions (e.g. the integral of sin⁡x\sin x)... to be honest, the whole notion of an elementary function isn't something you need to worry about at A-level (and aren't you doing GCSEs? In which case you really won't need to worry about them for a long time).


Indeed this is an unusual function to see at A-level maybe it's an "extra credit" question?
Original post by im so academic
So an elementary function is basically any function that's not an integral or a differential equation?


Well, no, the definition of an elementary function is somewhat arbitrary. Indeed, since the integral of e^(-x^2) (and its various linear transformations) occurs so frequently, it's convenient to define the so-called "error function" as in http://en.wikipedia.org/wiki/Error_function. Now, should we say that erf is an elementary function (why not?), then ∫exp⁡(−x2)\int \exp(-x^2) is expressible in terms of elementary functions. So why the list of elementary functions is restricted to exponentials, logarithms, powers and roots is mere convention, that these functions are prominent and "simple" enough to be considered elementary.
Reply 15
Avatar for Calrik
Calrik
Original post by Glutamic Acid
Well, no, the definition of an elementary function is somewhat arbitrary. Indeed, since the integral of e^(-x^2) (and its various linear transformations) occurs so frequently, it's convenient to define the so-called "error function" as in http://en.wikipedia.org/wiki/Error_function. Now, should we say that erf is an elementary function (why not?), then ∫exp⁡(−x2)\int \exp(-x^2) is expressible in terms of elementary functions. So why the list of elementary functions is restricted to exponentials, logarithms, powers and roots is mere convention, that these functions are prominent and "simple" enough to be considered elementary.


Tough thing to define, but anything that is linear, converges and can be expressed between limits is an elementary function. But of course that is open to discussion. Best thing to do is just accept that in maths it isn't an exact definition per se.
(edited 13 years ago)

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