Integrate e^x^2: Solving the Antiderivative & Steps | Help Me!

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In summary, the conversation discusses the evaluation of the integral of e^{x^2} and the methods that can be used to solve it. It is mentioned that while there is no antiderivative of e^{x^2} among elementary functions, it can be solved using a power series or by converting it to polar coordinates. The conversation also touches on finding the partial derivative of a function involving e^{x^2} and the importance of stating the entire question when seeking help with a problem.
  • #1
mdnazmulh
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there is no antiderivative of e[tex]^{x^{2}}[/tex] . So, I am puzzled how to evaluate the following integration. help me

[tex]\int e^{x^{2}} dx[/tex]

show me the steps if you have solved it.
 
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  • #2
Just because [itex]e^{x^2}[/tex] has no antiderivative among the elementary functions does not mean that it has no antiderivative, period. You can solve it as a power series.
 
  • #3
Multiply by [tex]e^{y^2}[/tex], then convert to polar coordinates, and take the sqrt at the end. Thats the usual ansatz for this one.
 
  • #4
D H said:
Just because [itex]e^{x^2}[/tex] has no antiderivative among the elementary functions does not mean that it has no antiderivative, period. You can solve it as a power series.

Yes, that' s right. First you should find the Taylor series representation of e^x. Then, in that series, you should replace x with x^2. Lastly, you should integrate it. It is simple, isn't it?

BTW, Integral of series is same with elemantary functions.
 
  • #5
AstroRoyale said:
Multiply by [tex]e^{y^2}[/tex], then convert to polar coordinates, and take the sqrt at the end. Thats the usual ansatz for this one.

ah, but that only works for the complete integral from 0 to ∞, doesn't it? :smile:
 
  • #6
actually, it's a problem from 'Calculus' by Anton Bivens Davis, 7th Edition. Chapter 14, Exercise no. 14.3, problem no. 63.

The whole problem states to find the partial derivative of f(x,y) with respect to x,

where, f(x,y)= [tex]\int ^{y}_{x} e^{t^{2}} dt[/tex]

Tomorrow I have an exam and that problem has strong chance to be given in exam. I have a soft copy of solution of Anton's Book. Unfortunately, in that solution only final answer partial derivative of f(x,y) with respect to x is given, f[tex]_{x}[/tex](x,y) = e[tex]^{x^{2}}[/tex]
How the answer came, it didn't show it.
Don't tell me to apply Taylor/Maclaurin's Law for e^x here BECAUSE answer then differs from the solution.
Many of you are expert in mathematics. So help me in this regard very quickly. Thanks in advance
 
  • #7
mdnazmulh said:
The whole problem states to find the partial derivative of f(x,y) with respect to x,

where, f(x,y)= [tex]\int ^{y}_{x} e^{t^{2}} dt[/tex]

Well, as D H indicated, it does have an antiderivative, so let's call it g(t).

Then ∂f/∂x = (∂/∂x)(g(y) - g(x))) = … ? :smile:

(btw, this thread show very clearly the importance of stating the whole question in the first place.)
 
  • #8
AstroRoyale said:
Multiply by [tex]e^{y^2}[/tex], then convert to polar coordinates, and take the sqrt at the end. Thats the usual ansatz for this one.
That only works when extending out to infinity or zero. If you try setting actual x and y bounds on the region R, you end up integrating e^(sec[x]^2) and another one just like that. It doesn't simplify, and if you want to integrate from zero to infinity, I'll tell you right now that the answer is infinity. You could expand the integrand as a power series and use termwise integration, however. That might not be terribly helpful in expressing the answer, but it does work.
 
  • #9
tiny-tim said:
ah, but that only works for the complete integral from 0 to ∞, doesn't it? :smile:

And if it's e^(-x^2). The integral of e^(x^2) from 0 to infinity is obviously divergent.
 
  • #10
mdnazmulh said:
The whole problem states to find the partial derivative of f(x,y) with respect to x,

where, f(x,y)= [tex]\int ^{y}_{x} e^{t^{2}} dt[/tex]

Well, this is an entirely different problem. There is no need to calculate the integral of [itex]\exp(t^2)[/itex] with respect to [itex]t[/itex]. Just use the fundamental theory of calculus.
 
  • #11
… to boldly go …

nicksauce said:
tiny-tim said:
ah, but that only works for the complete integral from 0 to ∞, doesn't it? :smile:

And if it's e^(-x^2). The integral of e^(x^2) from 0 to infinity is obviously divergent.

:wink: … still works … ! :wink:
 

1. What is the purpose of integrating e^x^2?

The purpose of integrating e^x^2 is to find the area under the curve of the function e^x^2. This can be useful in various applications such as calculating probabilities in statistics or solving differential equations in physics or engineering.

2. How do I solve the integral of e^x^2?

To solve the integral of e^x^2, you can use the substitution method or integration by parts. The substitution method involves substituting u = x^2 and du = 2x dx, while integration by parts involves splitting the integral into two parts and using the formula ∫udv = uv - ∫vdu.

3. Can I use a calculator to integrate e^x^2?

Yes, you can use a calculator or a computer program to solve the integral of e^x^2. However, it is important to understand the concepts and techniques behind integration in order to use these tools effectively.

4. Are there any special cases when integrating e^x^2?

Yes, when integrating e^x^2, there are two special cases: when the limits of integration are from -∞ to ∞ and when the limits of integration are from 0 to ∞. In these cases, the integral can be solved using special techniques such as the error function and the Gamma function.

5. How is the integral of e^x^2 related to the Gaussian function?

The integral of e^x^2 is closely related to the Gaussian function, also known as the normal distribution. The Gaussian function is often used to model natural phenomena, and the integral of e^x^2 is used to calculate the probability of a certain value occurring within a given range in the Gaussian distribution.

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