# Indefinite and definite integral of e^sin(x) dx

1. Apr 15, 2015

### Emmanuel_Euler

Look to this indefinite integral →∫e^(sin(x))dx
Antiderivative or integral could not be found.and impossible to solve.

Look to this definite integral
∫e^(sin(x))dx (Upper bound is π and Lower bound is zero)=??

my question is : can we find any solution for this integral (definite integral) ??

Last edited: Apr 15, 2015
2. Apr 15, 2015

### axmls

There is no closed-form solution for the antiderivative, but we can still approximate the definite integral.

3. Apr 15, 2015

### pasmith

It may be possible to use contour integration to find an analytic value for $\int_0^\pi e^{\sin x}\,dx$.

4. Dec 1, 2016

### Cgty

Assume that we have a solution like that int(y dy)=int(e^sinx dx). It is clear we must find y^2/2=int(e^sinx dx). In order to equality, int[ln(y) dy]=int(sinx dx). Due to int(lny dy) is equal to y(lny-1); y(lny-1)=-cosx+c and y=[-cosx+c]/[lny-1]. We need to find y^2/2 therefore, y^2/2=[(cosx+c)/(lny-1)]^2/2. This is the solution of int(e^sinx dx) and we have a non-linear euation.

5. Dec 3, 2016

### lurflurf

lets consider
$$\frac{1}{\pi}\int_0^\pi\!e^{\sin(x)}\,\mathrm{d}x$$
I flipped through some books and did not find much on that, but I did find that
$$\frac{1}{\pi}\int_0^\pi\!e^{\cos(x)}\,\mathrm{d}x=\operatorname{I}_0(1)$$
I is the modified Bessel function of the first kind.
Also we know that
$$\operatorname{I}_0(1)=\frac{1}{\pi}\int_0^\pi\!\cosh(\sin(x))\,\mathrm{d}x\\ \operatorname{I}_0(1)\sim1.26606587775201$$
http://people.math.sfu.ca/~cbm/aands/page_376.htm
and
$$\operatorname{L}_0(1)=\frac{1}{\pi}\int_0^\pi\!\sinh(\sin(x))\,\mathrm{d}x\\ \operatorname{L}_0(1)\sim0.710243185937891$$
L is the Modified Struve Function
http://people.math.sfu.ca/~cbm/aands/page_498.htm
so
$$\frac{1}{\pi}\int_0^\pi\!e^{\sin(x)}\,\mathrm{d}x=\operatorname{I}_0(1)+\operatorname{L}_0(1)\sim1.97630906368990$$

6. Jan 11, 2017

### Emmanuel_Euler

can you give me the name of the books please, because i need them and thank you so much for help

7. Jan 11, 2017

### lurflurf

I found that in the famous Handbook of Mathematical Functions edited by M. Abramowitz and I. A. Stegun a "work for hire performed for the US Government" thus freely available.
For example here
http://people.math.sfu.ca/~cbm/aands/toc.htm
It is also of course available in print if you prefer.

8. Jan 11, 2017

### Emmanuel_Euler

thank you so much for help.....