# INTEGRAL{ e^cos(x) dx }

1. May 20, 2012

### Si14

1. The problem statement, all variables and given/known data

I want to solve this integral without using series expansion. The answer should be in a closed form. I wonder if this is possible?

2. Relevant equations

3. The attempt at a solution
I used numerical methods and was able to solve it numerically for a given interval. However, I need to solve it without using numerical methods and without using series expansion.

Thanks.

2. May 20, 2012

### sharks

Try using substitution: Let y = cos x

3. May 20, 2012

### Si14

sharks, it seems that it gets more complicated!
I stuck immediately after substitution.

4. May 20, 2012

### Ray Vickson

I suspect it is not doable in terms of elementary functions. Neither Wolfram Alpha nor Maple 11 can find non-series expressions for the indefinite integral. You might try converting it so some known special (but non-elementary) function, perhaps by using integration by parts and/or substitution methods.

RGV

5. May 20, 2012

### Dickfore

The primitive function is not elementary. Do you want the definite or the indefinite integral? Because, if you look at the integral representation of the Bessel function, you might find a similarity.

6. May 20, 2012

### sharks

$$\int e^y.\frac{-1}{\sqrt{1-y^2}}dy$$Then, use integration by parts.

7. May 20, 2012

### Si14

I wonder if it is possible to solve the indefinite integral? I assume the answer to the definite one should be close to the answer I get with numerical methods.

8. May 20, 2012

### Dickfore

As I said, the primitive function is not elementary
Wolfram Alpha

9. May 20, 2012

### Si14

I checked wolframalpha. However, it gives a series expansion which I can not use.
I wonder if the integration by parts suggested by sharks is doable?

10. May 20, 2012

### Dickfore

You obviouly don't know what primitive or elementary means. Did you see the first sentence mentioned in wolfram alpha? It tells an even more stringent condition, in terms of "standard mathematical functions", which includes some non-elementary functions (including Bessel functions).

11. May 20, 2012

### sharks

Try $u=e^y$ and $dv=\frac{-1}{\sqrt{1-y^2}}$
Then, $du=e^y$ and $v=\cos^{-1}y$

12. May 20, 2012

### Ray Vickson

It won't get you anywhere; the integral is non-elementary, and no amount of manipulation will change that fact. However, you might try to re-express the indefinite integral in terms of some already-defined special functions (as already suggested).

RGV