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

[ns5032]

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!

 

[FrogPad]

I'm pretty sure you need to solve this numerically. Is that acceptable for the problem?

 

[olgranpappy]

It's very amusing to consider instead the function
$$e^{-t/5}$$
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
$$\int_0^{\infty} e^{(-t/5)}=5$$
whereas
$$\int_0^{\infty}e^{(Cos[t]-t)/5}\approx 5.09$$