Using Cauchy integral formula to compute real integral?

[Mixer]

Homework Statement

Compute the following integral around the path S using Cauchys integral formula for derivatives:

\inte^z / z²

Integral path S is a basic circle around origin.

Then, use the result to compute the following integral

\int e^{cos (x)} cos(sin (x) - x) dx from 0 to ∏

Homework Equations

Cauchy integral formula: http://en.wikipedia.org/wiki/Cauchy's_integral_formula

The Attempt at a Solution

I was able to compute the first part and got 4∏i. But I'm stuck in second part. How I'm supposed to use the result I got in first part to compute the second? Any hint for me?

 

[jackmell]

Homework Statement

Compute the following integral around the path S using Cauchys integral formula for derivatives:

\inte^z / z²

Integral path S is a basic circle around origin.

Then, use the result to compute the following integral

\int e^{cos (x)} cos(sin (x) - x) dx from 0 to ∏

Homework Equations

Cauchy integral formula: http://en.wikipedia.org/wiki/Cauchy's_integral_formula

The Attempt at a Solution

I was able to compute the first part and got 4∏i. But I'm stuck in second part. How I'm supposed to use the result I got in first part to compute the second? Any hint for me?

Well, you know \cos(a-b)=\cos(a)\cos(b)+\sin(a)\sin(b) and you got \cos(a-b) in there. Ok, keep that in mind.

Now, what do you get when you let z=e^{it} in the expression:

\int \frac{e^z}{z^2}dz

and then split the integral into a real and imaginary part?

 

[Gib Z]

Hint 1: \int_{\gamma} f(z) dz = \int^b_a f( \gamma(t) ) \gamma'(t) dt.

Hint 2: \cos(\sin x - x) = \operatorname{Re} e^{i\sin x - ix}.

 

[Mixer]

Thank you very much!

I actually had a typo in my first post. So the integral I'm supposed to compute is

\inte^{2 cos (t)}cos(sin(t) - t) dt from 0 to ∏

I was, however, able to compute the integral you gave me to this kind of form (this is an imaginary part of the integral:

\inte^{2 cos (t)} cos (sin(t) - t) dt from 0 to 2∏

is equal to 4∏ (the result I got in first part)

Now, how can I get the correct value when integrating from 0 to ∏ ?