Question

Integrate

Answer 1

\[\Huge \int\limits\limits_0^{2\pi} e^{\cos \theta} \cos (\sin \theta) d \theta\]

Answer 2

@dan815

Answer 3

i want to complexify

Answer 4

go ahead

Answer 5

integral of e^cos(theta)*e^(isintheta)
integral e^(costheta+isintheta)
integrate and take real part?

Answer 6

what just happened? where did cos(sin theta) go?

Answer 7

i complexified it

Answer 8

how do we complexify?

Answer 9

e^(e^itheta) :/

Answer 10

e^(isintheta) = cos(sintheta) + isin(sintheta)

Answer 11

ohh i see :)

Answer 12

so we get e^(cos x+ i sin x) * (-sinx+i cos x)
take the real part of that and then evaluate the bounds

Answer 13

dan

\(\large \int e^{\cos x} dx \ne e^{\cos x} (-\sin x )\)

right ?

Answer 14

yeah that statement is right

Answer 15

true xD

Answer 16

we shud make it into another complex

Answer 17

e^(costheta+isintheta) = e^(e^itheta)

Answer 18

yeah but the indefinite integral doesn't seem to evaluate to known functions

http://www.wolframalpha.com/input/?i=%5Cint+e%5E%28cos+%5Ctheta%29cos%28%5Csin+%5Ctheta%29

Answer 19

apparently it evaluates to 2pi that integral :O, but that is with the unreal part

Answer 20

what does this become in taylor expansion?