Question

Integrate please

Answer 1

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

Answer 2

I tried to use the property that \[\LARGE \int\limits_0^{2a} f(x)dx=2 \int\limits_0^a f(x)dx\]
if it is an even function..so it is..so I ended up with
\[\LARGE 2\int\limits_{0}^{\pi} e^{\cos \theta} \cos(\sin \theta)\]
Using property again,
\[\LARGE 2 \int\limits_{0}^{\pi} e^{-\cos \theta} \cos(\sin \theta)\]
now we can add the above 2 but I don't think that's gonna help :( just an attempt.

Answer 3

Open study won't show the eqautions well.

Answer 4

http://i.imgur.com/XqbtxG2.png

Answer 5

i have a hunch this cant be done ...i know e^cos doesnt have an anti-derivative

Answer 6

the answer is 2pi confirmed by WF

Answer 7

sorry i meant there doesnt exist an indefinite integral with elementary functions

Answer 8

@Loser66 that is the indefinite integral which is not nice, but the definite integral from 0 to 2pi is indeed 2pi as its just the area under the curve

Answer 9

@INeedhelpPlease?
not sure what you want, all you can do is approximate area under curve and show it converging to 2pi

Answer 10

@hartnn

Answer 11

That would be a messy integral for sure, what level calculus are you in?Just to point something out, you can't pull out a constant from your limits of integration. That concept would be true if the 2 was in front of the function, but you would be changing the integral by removing it from the limits.

Answer 12

I can.

Answer 13

write cos (sin t) as e^(i sin t)+ e^(-i sin t) /2

Answer 14

what did you try ?

Answer 15

got e^(cos t + isin t) + e^(cos t - i sin t)
???
thrn put
z = cost +i sin t
1/z = cos t - i sin t

e^z + e^1/z

Answer 16

hmm, actually e^1/z will not be integrable

Answer 17

its 0 to pi, right ?
so,
e^(cos t - i sin t ) ----> e^ (- cos t - i sin t ) = e^ -(cos t + i sin t) = e^-z
now thats integrable

Answer 18

you know i have used this property there, right ?
integral a to b f(x) dx = integral a to b f(a+b-x) dx

Answer 19

kuch samaj aaya ?

Answer 20

arre ek min :|

Answer 21

maine bhi to use kiya wahi a+b-x but not in starting after using even one?

Answer 22

i wrote my attemmpt in beginning btw

Answer 23

what ? didn't get u ...

yes, you used it there, but it didn't help much
but it helped for e^1/z to get e^-z instead

Answer 24

2pi aa raha hai?