Evaluate INT (5+cosx)e^-x from 0 to infinity

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In summary, the integral of 1/ex from 0 to infinity converges to 1. However, it is not clear how to prove this convergence, as Dick's simpler suggestion would suffice.
  • #1
IntegrateMe
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ok so i got (5+cos x)/ex and i compared that with 1/ex

(they're both going from 0 to infinity).

Turns out that the integral of 1/ex from 0 to infinity converges to 1. But i don't know how to prove that our original function converges as well (which is the answer). Anyone care to help?
 
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  • #2
Split the integral into: [tex]5\int_0^\infty{e^{-x}}dx+\int_0^\infty{cosxe^{-x}}dx[/tex]

and since you know the first one is equal to 5, just use integration by parts to find the second and then add 5 to it.
 
  • #3
You could also compare your original integral with 6/e^x. Which is larger?
 
  • #4
If you're just trying to prove that the integral converges, the fact that [itex]\int_0^{\infty}\exp{(-x)}\mathrm{d}x[/itex] converges is enough to prove that the second integral in Mentallic's post converges. For example, [itex]|\cos{(x)}| \leq 1[/itex] from which it follows that [itex]\exp{(-x)}|\cos{(x)}| = |\exp{(-x)}\cos{(x)}| \leq \exp{(-x)}[/itex] and then using this inequality we can reduce the convergence of the second integral to mere trivialities. However, if you actually need to evaluate the integral, Mentallic's method is definitely the best way to go.

Edit: I would follow Dick's advice since his suggestion is considerably simpler than mine.
 
  • #5
jgens said:
If you're just trying to prove that the integral converges, the fact that [itex]\int_0^{\infty}\exp{(-x)}\mathrm{d}x[/itex] converges is enough to prove that the second integral in Mentallic's post converges. For example, [itex]|\cos{(x)}| \leq 1[/itex] from which it follows that [itex]\exp{(-x)}|\cos{(x)}| = |\exp{(-x)}\cos{(x)}| \leq \exp{(-x)}[/itex] and then using this inequality we can reduce the convergence of the second integral to mere trivialities. However, if you actually need to evaluate the integral, Mentallic's method is definitely the best way to go.

Edit: I would follow Dick's advice since his suggestion is considerably simpler than mine.

Nah, it's not that much simpler. Mine only works because (5+cos(x)) is nonnegative. If it were (1+cos(x)) IntegrateMe would definitely want to use your comparison.
 
  • #6
First break the integrand into twp parts as shown by Mentallic.You can reduce the pain of integration by writing cos(x) as Re[exp(ix)];next solve the integration,which is now trivial.Then,take the real part of the resulting indefinite integral and put the limits.The lower limit gives you finite value and the upper limit gives you zero for the e^-x factor.
 

What is the purpose of evaluating INT (5+cosx)e^-x from 0 to infinity?

The purpose of evaluating this integral is to determine the area under the curve of the function (5+cosx)e^-x from 0 to infinity. This can provide information about the behavior and properties of the function.

Is this integral solvable analytically?

Yes, this integral can be solved analytically using integration techniques such as integration by parts or substitution.

What is the value of this integral?

The value of this integral is equal to 5. This can be found by using integration techniques to evaluate the integral.

How does the value of the integral change as the upper limit approaches infinity?

As the upper limit approaches infinity, the value of the integral approaches 5. This is because the function (5+cosx)e^-x approaches 0 as x approaches infinity, making the area under the curve smaller and closer to 5.

What is the significance of the function (5+cosx)e^-x in this integral?

The function (5+cosx)e^-x is the integrand in this integral and its behavior determines the shape and area under the curve. It is a combination of a polynomial and exponential function, which can have interesting properties and can be useful in modeling various phenomena in science and mathematics.

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