Solving an Integral Using Euler's Formula

[cragar]

I was trying to do this integral, this is not a homework question.

<br /> <br /> \int_{0}^{2\pi}e^{e^{ix}}dx<br />
I tried writing the e^(ix) part using eulers formula .
anyone have any other suggestions

 

[dextercioby]

I don't see other options than to express it using Euler's formula. As for the final result itself, the Mathematica software from wolframalpha.com returns a numerical approximation, not a combination of special functions.

 

[AlephZero]

Let u = e^{ix}
du = iu\,dx
The integral then becomes

-i \int \frac{e^u}{u}du

See http://mathworld.wolfram.com/ExponentialIntegral.html for more.