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calculus and analysis - Interpolating an Antiderivative



I'd like to be able to make InterpolatingFunctions for antiderivatives of functions that can't be integrated symbolically. However, the following code returns several error messages:


FunctionInterpolation[Integrate[Sqrt[1 + x^3], {x, 0, t}], {t, 0, 10}]

Here are the messages:



Thread::tdlen: "Objects of unequal length in {-1.25,-0.416667,0.416667,1.25}^{} cannot be combined. "


Thread::tdlen: "Objects of unequal length in {0.223144 +3.14159\ I,-0.875469+3.14159\ I,-0.875469,0.223144}\ {}\n cannot be combined. "


Thread::tdlen: "Objects of unequal length in {-1.25,-0.416667,0.416667,1.25}\ {} cannot be combined."


General::stop: "Further output of Thread::tlden will be suppressed during this calculation."


FunctionInterpolation::nreal: Near t = 1.25`, the function did not evaluate to a real number.



FunctionInterpolation::nreal: Near t = 1.3277777777777777`, the function did not evaluate to a real number.



What's going on? Is there a simple way to make this work? Changing Integrate to NIntegrate doesn't help, though the error messages are different:



NIntegrate::nlim: x = t is not a valid limit of integration.


NIntegrate::nlim: x = t is not a valid limit of integration.


NIntegrate::nlim: x = t is not a valid limit of integration.


General::stop: "Further output of NIntegrate::nlim will be suppressed during this calculation. "




Answer




Use NDSolve


antiD = NDSolveValue[{f'[x] == Sqrt[1 + x^3], f[0] == 0}, f, {x, 0, 10}]

Example usage:


Plot[antiD[x], {x, 0, 10}]

Mathematica graphics



Alternatively...


This works because this function can be antidifferentiated (by Mathematica).



antiD = FunctionInterpolation[
 Evaluate @ Integrate[Sqrt[1 + x^3], {x, 0, t}, Assumptions -> 0 < t < 10],
 {t, 0, 10}]

or...


integral[t_?NumericQ] := NIntegrate[Sqrt[1 + x^3], {x, 0, t}];
FunctionInterpolation[integral[t], {t, 0, 10}]

FunctionInterpolation evaluated its argument on a symbolic t. The pattern test ?NumericQ prevents evaluation of NIntegrate until an actual number is substituted for t. See also What are the most common pitfalls awaiting new users?. Note also that this way does many evaluations of NIntegrate, whereas the NDSolve method does just one integration.


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