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

  1. Apr 15, 2015 #1
    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. jcsd
  3. Apr 15, 2015 #2
    There is no closed-form solution for the antiderivative, but we can still approximate the definite integral.
     
  4. Apr 15, 2015 #3

    pasmith

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    It may be possible to use contour integration to find an analytic value for [itex]\int_0^\pi e^{\sin x}\,dx[/itex].
     
  5. Dec 1, 2016 #4
    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.
     
  6. Dec 3, 2016 #5

    lurflurf

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    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
    $$\frac{1}{\pi}\int_0^\pi\!e^{\sin(x)}\,\mathrm{d}x=\operatorname{I}_0(1)+\operatorname{L}_0(1)\sim1.97630906368990$$
     
  7. Jan 11, 2017 #6
    can you give me the name of the books please, because i need them and thank you so much for help
     
  8. Jan 11, 2017 #7

    lurflurf

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    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.
     
  9. Jan 11, 2017 #8
    thank you so much for help.....
     
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