How Similar Are the Integrals of e^(cos(t)-t)/5 and e^(-t/5)?

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The discussion focuses on the numerical evaluation of the integral of the function e^[(cos(t)-t)/5]. Participants agree that numerical methods are acceptable for solving this integral. They compare it to the simpler integral of e^(-t/5), which evaluates to 5, while the integral of e^[(cos(t)-t)/5] is approximately 5.09. This indicates that the two integrals are closely related, despite the additional complexity in the first function.

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I'm in the middle of a LONG problem and came across a part where I need to do this integral:

integral of: e^[(cos(t)-t)/5]

Please help!
 
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I'm pretty sure you need to solve this numerically. Is that acceptable for the problem?
 
It's very amusing to consider instead the function
<br /> e^{-t/5}<br />
It looks almost just like your function but with a few less little "fine structure" wiggles.

The integrals of both functions should be rather similar. For example
<br /> \int_0^{\infty} e^{(-t/5)}=5<br />
whereas
<br /> \int_0^{\infty}e^{(Cos[t]-t)/5}\approx 5.09<br />
 

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