plotting - Slow plot for integral involving FractionalPart and sine

I have defined this function:

f[x_]:=1/(1+Sin[FractionalPart[x]])

Then I tried to plot it using this code

Plot[Integrate[f[t],{t,0,x}],{x,0,2}]

It finally plot it, but it takes about 4-5 minutes to do it. There is a way to speed up similar plots, that involve commands as FractionalPart?

EDIT: I observed that changing Integrate by NIntegrate the plot speed up a lot, but Im not sure how accurate is the numerical integration to trust in the result of the plot.

Anyway I will like to know if there are other approaches to this problem using the symbolic integration. Thank you.

Answer

You could use DSolve to calculate the symbolic integral over a finite interval.

ClearAll[f];
f[x_] := 1/(1 + Sin[FractionalPart[x]])

F = y /. First@DSolve[{y'[x] == f[x], y[0] == 0}, y, {x, 0, 2}]

Plot[F[x], {x, 0, 2}]

Mathematica graphics

It's reasonably fast:

Mathematica graphics

Addendum: There is a reason I thought to try this. Events were added to DSolve a couple of versions ago, and these are used in discontinuity processing. What I imagine happens is this. When you've got an integrand with a standard discontinuous function, DSolve will split up the intervals and feed simple, continuous integrands to Integrate[]. It then adds up the results and pieces them together. At least that's how it seems when this trick works. I'm not familiar with the internal implementation.

Comments

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