- #1
eyesontheball1
- 31
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Evaluate the following integral:
[itex]\int_0^{∞}[/itex] [itex]\frac{e^{-(x+x^{-1})}}{x}dx[/itex]
[itex]\int_0^{∞}[/itex] [itex]\frac{e^{-(x+x^{-1})}}{x}dx[/itex]
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Evaluate the following integral
- Thread starter eyesontheball1
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- Integral
In summary, the conversation is about evaluating the integral given by \int_0^{∞} \frac{e^{-(x+x^{-1})}}{x}dx, with the suggestion of expressing e as (1+1/x)^x and using the substitution u=1/x. However, it is pointed out that the integral does not converge and can be expressed in terms of a Bessel function.
- #2
joeblow
- 71
- 0
.227788.
(?)
- #3
eyesontheball1
- 31
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Evaluate the following integral (symbolically and not numerically), should've specified that.
- #4
joeblow
- 71
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My guess is that you express e = (1+1/x)^x then work with that.
- #5
Whovian
- 652
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joeblow said:
Since x is already a variable in the problem, I assume you mean ##e=\lim\limits_{n\to0}\left(\left(1+\dfrac1n \right)^n\right)##?
For some reason, I feel like some sort of substitution of ... wait a second ...
How about the substitution ##u=\dfrac1x##?
- #6
Millennial
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The integral given does not converge. Its antiderivative is [itex]\displaystyle e^{-1/x}\text{Ei}(-x)[/itex] where Ei is the exponential integral function. I think simply plugging in zero for x shows why it wouldn't converge.
- #7
- #8
eyesontheball1
- 31
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Thank you again JJacquelin! I can always count on you! :)
What is an integral?
An integral is a mathematical concept that represents the area under a curve in a graph. It is a fundamental tool in calculus and is used to solve many real-world problems.
How do you evaluate an integral?
To evaluate an integral, you need to use the fundamental theorem of calculus, which states that the integral of a function can be calculated by finding the antiderivative of that function and evaluating it at the upper and lower bounds of the integral. This can be done using various techniques such as substitution, integration by parts, or partial fraction decomposition.
What are the limits of integration?
The limits of integration are the upper and lower bounds of the integral, which define the range over which the function is being integrated. These limits can be constants, variables, or expressions.
Why is it important to evaluate an integral?
Evaluating integrals is important because it allows us to find exact solutions to real-world problems that involve continuous change. It is also a vital tool in many branches of mathematics and physics, such as calculating areas, volumes, and probabilities.
What is the difference between definite and indefinite integrals?
A definite integral has specific upper and lower bounds, while an indefinite integral does not. A definite integral represents a single numerical value, while an indefinite integral represents a family of functions that differ by a constant. In other words, a definite integral gives a specific answer, while an indefinite integral gives a general solution.
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