# How to include conditional statements in NIntegrate?

I have a function g which is the result of another integrated function f. This function (f) has some singularities for some input values. While integrating function f I want to tell NIntegrate to take some value say p at singularities.

f[x1_, x2_, t_, ra_] = Sin[x1]*ra +
ra*Cos[x2]*ArcTan[x1*ra + t*x2,
ra*t + x1*x2];


This function has singularities for ArcTan[0,0].

g[x1_, x2_, t_] :=
With[{ff = f[x1, x2, t, ra]},
NIntegrate[ff, {ra, 0, 10^(-3)},
Exclusions -> ArcTan[__, __] == 0]];


My actual code is about 300 lines long. The above problem might help me to solve my actual problem in Mathematica. Kindly anyone help me.

EDITED

I have function "f" inside this function I have many h1[ArcTan[exp1, exp2]] , h2[ArcTan[exp3, exp4]],....functions (may be more than 50 of ArcTan's are there). For some input values exp1, exp2, exp3, exp4, exp... goes to 0. I applied

f /. ArcTan[0, 0] -> p


at the end of the function. I was thinking When I when I substitute the input values inside f1 it will have ArcTan[0,0] and that gets replaced with "p".But it is not doing that. Is there a way to Replace ArcTan[0,0] with "p" .Then this f1 function is called for another function g where it is integrated.

• I have updated the answer taking into account your "EDITED" section. Oct 14, 2019 at 7:13

Please read carefully the Documentation pages for Exclusions and NIntegrate: your understanding of this option is wrong. It doesn't allow to specify a specific value at exclusions as you apparently think. NIntegrate is supposed to automatically handle singularities and all what you can do is to help it a bit by specifying them via this option. The correct usage in your case would be

Exclusions -> {x1*ra + t*x2 == 0 && ra*t + x1*x2 == 0}


If you really do not see a better way than providing a specific value for you function at singularity points (what is a bad idea in general) you could try to turn your objective function into a Piecewise or a black-box function with the specific value p for those points:

NIntegrate[Piecewise[{{p, x1*ra + t*x2 == 0 && ra*t + x1*x2 == 0}},
f[x1, x2, t, ra]], {ra, 0, 10^(-3)}]


or (if the previous doesn't work)

fBlackBox[x1_?NumericQ, x2_?NumericQ, t_?NumericQ, ra_?NumericQ] :=
Piecewise[{{p, x1*ra + t*x2 == 0 && ra*t + x1*x2 == 0}}, f[x1, x2, t, ra]];
NIntegrate[fBlackBox[x1, x2, t, ra], {ra, 0, 10^(-3)}]


You can also replace AcrTan with a Piecewise function having value p at the singular point:

arcTan[x_, y_] := Piecewise[{{p, x == 0 && y == 0}}, ArcTan[x, y]]
f[x1_, x2_, t_, ra_] := Sin[x1]*ra + ra*Cos[x2]*arcTan[x1*ra + t*x2, ra*t + x1*x2];


Of course, you can also define it via Condition but I'm not sure whether NIntegrate will handle such a definition correctly, so you probably should define f as a black-box function in this case as I do below:

arcTan[x_, y_] /; x == 0 && y == 0 := p
arcTan[x_, y_] := ArcTan[x, y]
fBlackBox[x1_?NumericQ, x2_?NumericQ, t_?NumericQ, ra_?NumericQ] :=
Sin[x1]*ra + ra*Cos[x2]*arcTan[x1*ra + t*x2, ra*t + x1*x2];

• hi, thanks for the reply. I tried this way it did not work though. f[x1_, x2_, t_, ra_] = Sin[x1]*ra + raCos[x2]*ArcTan[x1*ra + tx2, ra*t + x1*x2] /; ArcTan[,] == ArcTan[0,0] -> p; Is there a way to write it using Condition function Oct 13, 2019 at 16:47
• @GummalaNavneeth I'm sorry, but your code ArcTan[, ] == ArcTan[0, 0] just demonstrates that you don't understand even the basics of the Wolfram Language. You should learn the basics of the Wolfram Language. I recommend the book by Leonid Shifrin as a quick start. It won't take much of your time. Please learn it before asking new questions. Oct 13, 2019 at 18:28
• @GummalaNavneeth And Yes, it is possible to write it via Condition, although the ways I showed in the answer should be better. But in any case, without understanding of the basics you won't be able to go further with this language. Oct 13, 2019 at 18:42
• @Alexwy Popkov Sorry for the trouble caused. I will go through that book. Thanks Oct 13, 2019 at 20:46
• @Alexwy Popkov Thanks a lot Oct 14, 2019 at 9:51