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I am trying to integrate Abs|E^ $\sin \theta$| (the absolute value of Euler's constant to the power of sin(theta), and tried the code Integrate[Abs[E^Sin[x]], x] but it returned as \[Integral]E^Re[Sin[x]] \[DifferentialD]x

Could someone please show me how to correctly integrate the function? Thanks, and sorry in advance for the corrections you would have to make to this post!

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  • $\begingroup$ I am sure the antiderivative under consideration has no closed-form expression. Think of the function NIntegrate[Abs[E^Sin[t]], {t, 0, x}]. $\endgroup$
    – user64494
    Commented May 27, 2019 at 3:49
  • $\begingroup$ Or better, to make @user64494's point more clear: Plot[NIntegrate[RealAbs[E^Sin[x]], {x, -10, t}],{t, -10, 10}] $\endgroup$
    – b3m2a1
    Commented May 27, 2019 at 3:52
  • $\begingroup$ thanks for your clarification, @b3m2a1 $\endgroup$
    – wendy
    Commented May 27, 2019 at 9:38
  • $\begingroup$ thank you for your advice, @user64494 ! $\endgroup$
    – wendy
    Commented May 27, 2019 at 9:39

1 Answer 1

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Check this numerical treatment: 

g[t_] := Abs[E^Sin[t]]
nsol = Table[NIntegrate[g[t], {t, 0, x}], {x, -10, 10, .1}]; 
f[x_] = Interpolation[Thread@{Table[x, {x, -10, 10, .1}], nsol}, x]
Plot[f[x], {x, -1, 1}]

Now this f[x] can be used in further calculations!

enter image description here

Let's check the obtained interpolated function f(x) is correct or not!

 NIntegrate[g[t], {t, -10, 10}]  = 25.0755
 f[10] - f[-10]                  = 25.0755

Hope This helps!

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  • $\begingroup$ it does, @Sachin Kumar ! Thanks so much for ur help $\endgroup$
    – wendy
    Commented May 27, 2019 at 9:36
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    $\begingroup$ @wendy, Good that it has helped you, Pl accept the answer. $\endgroup$
    – Schrodinger
    Commented May 27, 2019 at 9:42
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    $\begingroup$ Better to use FunctionInterpolation. It'll be smarter about how it chooses points and the tests I've run suggest it's about four times as accurate as this method. Also it's faster. If you want to stick with Interpolation, at the very least it makes sense to create the grid of points and the interpolated values in the same Table call. $\endgroup$
    – b3m2a1
    Commented May 27, 2019 at 16:26

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