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# Integrating e^f(x) - is this possible? Tweet

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1. Integrating e^f(x) - is this possible?
Hey,
I'd like to integrate

I'm not sure if this is possible though. I haven't been set this question, I just randomly thought of it.

My first guess of the integral was:

i.e. that e^f(x) integrates -> 1/f'(x) * e^f(x)

However, I know this is wrong, because when differentiating my integral, I'd have to use the quotient rule and the result is not my original.

I also tried it on Mathematica on the Wolfram website, and got something weird containing an "erfi", which I haven't encountered before.

Any thoughts? Have I missed something? Thanks
2. Re: Integrating e^f(x) - is this possible?
I suppose it depends what you mean by "possible". Let me explain.

The reason why your method doesn't work is because you're integrating by substitution wrong. What you're saying is

(ignore the constant of integration for now). Rearranging this gives

But the rule of integration by substitution (which is the 'opposite' of the chain rule) says

These two integrals are only equal when is constant. In this case, , which isn't constant, and so .

In fact, the your integral cannot be expressed in terms of elementary functions, and so instead it is expressed in terms of the (imaginary) 'error function'. The error function is defined by -- note how this function depends on the limit of the integral -- and this is not what we call an 'elementary function' (for reasons I won't go into). The imaginary error function is , where , which corresponds to the fact that your coefficient of is positive.

So to answer your question a bit more briefly: that cannot be integrated in terms of elementary functions (which I guess means 'can't be integrated' for A-level).

For more, see here.

What's the problem you've been set?
Last edited by nuodai; 29-07-2012 at 20:02.
3. Re: Integrating e^f(x) - is this possible?
No it's not possible to integrate this and have an answer in terms of elementary functions
4. Re: Integrating e^f(x) - is this possible?
(Original post by geekly)
Hey,
I'd like to integrate

I'm not sure if this is possible though. I haven't been set this question, I just randomly thought of it.

My first guess of the integral was:

i.e. that e^f(x) integrates -> 1/f'(x) * e^f(x)

However, I know this is wrong, because when differentiating my integral, I'd have to use the quotient rule and the result is not my original.

I also tried it on Mathematica on the Wolfram website, and got something weird containing an "erfi", which I haven't encountered before.

Any thoughts? Have I missed something? Thanks
The reverse differentiation method that you've tried to use usually only works for linear functions. Most of the time, it's best to use a substitution if you're not confident that reverse differentiation will work.

And if you notice a "weird" function that you haven't seen before in the integral when using e.g. Wolfram Alpha then in A Level terms, this means that you can't integrate it.
Last edited by notnek; 29-07-2012 at 20:01.
5. Re: Integrating e^f(x) - is this possible?
Not possible to integrate in terms of elementary functions.
6. Re: Integrating e^f(x) - is this possible?
Ahh. Okay, thanks everyone
7. Re: Integrating e^f(x) - is this possible?
y = e^(2(x + 3/2)^2 + 7/2) = e^7/2 * e(2(x + 3/2)^2 which can be integrated with the erf(x) function.
8. Re: Integrating e^f(x) - is this possible?
(Original post by nuodai)
What's the problem you've been set?
School's finished and I haven't been set anything, I was just doing an old maths paper for some revision . I was asked to differentiate a similar function to the one above, and I started wondering what would happen if I wanted to integrate it.

Thanks for the link, I will print and add to my notes when I get home
9. Re: Integrating e^f(x) - is this possible?
(Original post by geekly)
Hey,
I'd like to integrate

I'm not sure if this is possible though. I haven't been set this question, I just randomly thought of it.

My first guess of the integral was:

i.e. that e^f(x) integrates -> 1/f'(x) * e^f(x)

However, I know this is wrong, because when differentiating my integral, I'd have to use the quotient rule and the result is not my original.

I also tried it on Mathematica on the Wolfram website, and got something weird containing an "erfi", which I haven't encountered before.

Any thoughts? Have I missed something? Thanks
I followed a process of completing the square. It came down to a constant multiplying the integral of e^2u^2. I substituted v for u and proceeded to get a form that was subject to integration by parts.
10. Re: Integrating e^f(x) - is this possible?
(Original post by aurao2003)
I followed a process of completing the square. It came down to a constant multiplying the integral of e^2u^2. I substituted v for u and proceeded to get a form that was subject to integration by parts.
You must have gone wrong somewhere; it's impossible to find an expression for in terms of elementary functions, no matter how much substitution and parts you apply to it.
11. Re: Integrating e^f(x) - is this possible?
(Original post by nuodai)
You must have gone wrong somewhere; it's impossible to find an expression for in terms of elementary functions, no matter how much substitution and parts you apply to it.
Make 2u^2 a constant.
12. Re: Integrating e^f(x) - is this possible?
(Original post by aurao2003)
Make 2u^2 a constant.
What do you mean?
13. Re: Integrating e^f(x) - is this possible?
How can you make it constant when that is the variable that you are integrating with respect to?
14. Re: Integrating e^f(x) - is this possible?
(Original post by nuodai)
What do you mean?
You could say 2u^2 = k. (Sorry meant to say a variable)
15. Re: Integrating e^f(x) - is this possible?
(Original post by aurao2003)
You could say 2u^2 = k. (Sorry meant to say a variable)
You have already been told that this integral cannot be evaluated in terms of elementary functions.
16. Re: Integrating e^f(x) - is this possible?
(Original post by aurao2003)
You could say 2u^2 = k. (Sorry meant to say a variable)
Yes but when you apply this substitution you would get another integral which cannot be expressed in terms of elementary functions
17. Re: Integrating e^f(x) - is this possible?
(Original post by aurao2003)
You could say 2u^2 = k. (Sorry meant to say a variable)
That transforms the integral to become

for which we have the same problem.
Last edited by nuodai; 06-08-2012 at 23:44. Reason: Erp... mistake fix'd
18. Re: Integrating e^f(x) - is this possible?
(Original post by nuodai)
That transforms the integral to become

for which we have the same problem.
I concur.