Integral of e^cosx: Answers Sought

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In summary, the conversation discusses the integral of e^cosx and the possibility of expressing it in terms of elementary functions. There is no definite answer and some have attempted using integration by parts. The function is periodic and has no closed form. The conversation also touches on the limits of e^cosx and e^-cosx as x approaches infinity, with the conclusion that these functions have no limits or are undefined at infinity.
  • #1
chwala
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Homework Statement
Find the indefinite integral of ##e^{\cos x}dx##
Relevant Equations
integration
I just came across this and it seems we do not have a definite answer...there are those who have attempted using integration by parts; see link below...i am aware that ##\cos x## has no closed form...same applies to the exponential function.

https://math.stackexchange.com/questions/2468863/what-is-the-integral-of-e-cos-x

would appreciate insight...

cheers!
 
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  • #2
y=e^(cos x) is a periodic function as we see in https://www.wolframalpha.com/input?i=graph+of+y=e^(cos+x) .
[tex]\int_0^{2\pi} e^{\cos x}dx := 2\pi a \approx 7.95493[/tex]
[tex] \int_0^{2\pi} (e^{\cos x}-a) dx = 0 [/tex]
This modified integral is periodic, i.e.,
[tex] \int_0^{X_1} (e^{\cos x}-a) dx = \int_0^{X_2} (e^{\cos x}-a) dx [/tex]
where
[tex] X_2=X_1-2n\pi ,0<X_2<2\pi[/tex]
Any of the modified integral is reduced to the integral in region ##[0,2\pi]##. Surely there exists the integral but I do not expect that it can be expressed by ordinary functions.

[tex]\int_{X_1}^{X_2} e^{\cos x}dx=I(x_2)-I(x_1)+a(X_2-X_1)[/tex]
where
[tex]x_1=X_1-2n_1\pi, 0<x_1<2\pi[/tex]
[tex]x_2=X_2-2n_2\pi, 0<x_2<2\pi[/tex]
[tex]I(x):=\int _0^x (e^{\cos t}-a)dt,\ 0<x<2\pi [/tex]
 
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  • #3
chwala said:
I will amend the question my bad ; didn't get the English correctly...I meant being expressed as a function.

I think perhaps you are looking for "expressed in terms of elementary functions"; it is trivial, but uninformative, to define [tex]
F(t) = \int_0^t e^{\cos u}\,du.[/tex]
 
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  • #4
I think you can try to have a infinite sum expanding by Taylor the exponential:

## \int e^{\cos(x)}dx=\int 1+\cos{x}+\frac{\cos^2{x}}{2!}+\frac{\cos^3{x}}{3!} dx ##

now by linearity:

## \int e^{\cos(x)}dx=x+\sin{x}+\int\frac{\cos^2{x}}{2!} dx+\int \frac{\cos^3{x}}{3!} dx + ...##

If you have a closed form for ##\int \cos^n{x}dx## I think you can find an expansion for ##\int e^{\cos{x}}dx##.

Ssnow
 
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  • #5
Ssnow said:
I think you can try to have a infinite sum expanding by Taylor the exponential:

∫ecos⁡(x)dx=∫1+cos⁡x+cos2⁡x2!+cos3⁡x3!dx

now by linearity:

∫ecos⁡(x)dx=x+sin⁡x+∫cos2⁡x2!dx+∫cos3⁡x3!dx+...

If you have a closed form for ∫cosn⁡xdx I think you can find an expansion for ∫ecos⁡xdx.

Ssnow
we may do further reduction making use of
1676259577561.png

from https://socratic.org/questions/how-do-you-find-the-integral-of-cos-n-x .
I am not patient and see it messy.
 
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  • #6
interesting ...how would we attempt to find then the limits of ##e^{-\cos x}## and ##e^{cos x}## as ##x## tends to infinity? my interest is on the approach, i can tell from the graph that the limits tend to ##±∞##.
 
  • #7
They are periodical functions as well as cos x is. They have no limits for x=##\pm \infty##.
1676290081170.png


[tex]e^{-\cos x}=e^{\cos(x+\pi)}[/tex]
1676290246239.png
 
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  • #8
anuttarasammyak said:
They are periodical functions as well as cos x is. They have no limits for x=##\pm \infty##.View attachment 322205

[tex]e^{-\cos x}=e^{\cos(x+\pi)}[/tex]
View attachment 322206
I need to look at this, can we say that it is bounded ...supremum, infimum? need to check on this man!
 
  • #9
[tex]e^{-1} \le e^{\cos x} \le e[/tex]
 
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  • #10
anuttarasammyak said:
[tex]e^{-1} \le e^{\cos x} \le e[/tex]
Nice, yes, If ##y=\cos x## then we have the maximum at ##y=1## and minimum at ##y=-1##... then how comes that this function has no limit? given
##\lim_{x \rightarrow +\infty} {e^{cos x}}##

am i missing something here...
 
  • #12
By "no limits", we usually mean "it diverges". So what is the limit of sin x when tends to infinity? It does not exist. We say sin x has no limit to infinity, or that at infinity sin x is undefined. Just a matter of wording, mathematicians once they write a formula they know exactly what it means, irrespective of which exact words are suited best to describe the formula.

"So a function/sequence/series does not converge to a limit, i.e. it diverges, or its limit does not exist". This should be clear. Let us not turn mathematics into semantics, a discipline of linguistics.
 
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1. What is the integral of e^cosx?

The integral of e^cosx is a mathematical expression that represents the area under the curve of the function e^cosx. It is a fundamental concept in calculus and is used to solve a variety of real-world problems.

2. How do you solve the integral of e^cosx?

To solve the integral of e^cosx, you can use the substitution method or integration by parts. The substitution method involves substituting u = cosx and du = -sinx dx, while integration by parts involves breaking down the integral into simpler parts and applying the product rule.

3. Is the integral of e^cosx a difficult concept to understand?

The concept of the integral of e^cosx may seem challenging at first, but with practice and understanding of the underlying principles, it can be easily grasped. It is important to have a strong foundation in basic calculus concepts before attempting to understand the integral of e^cosx.

4. What is the significance of the integral of e^cosx in real-world applications?

The integral of e^cosx has many real-world applications, such as in physics, engineering, and economics. It is used to calculate the area under curves, which is important in understanding rates of change, optimization, and probability distributions.

5. Are there any shortcuts or tricks to solve the integral of e^cosx?

While there are no specific shortcuts or tricks to solve the integral of e^cosx, having a good understanding of the properties of exponential and trigonometric functions can make the process easier. It is also helpful to practice solving various types of integrals to develop a better intuition for solving the integral of e^cosx.

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