# Calculate a numerical integration with NIntegrate?

GROUPS:
 Dear all, I'm struggling calculating a numerical integral. Here you can find attached (and also written in the post) two notebooks: R1 (where I used Integrate) and R2 (where I used Nintegrate). I think that R1 does not end the notebook evaluation because the integral is probably not solvable analytically, while R2 gives me a warning about the convergence and a wrong final result.What I'm after The result for the integrals (variable R) should be 0.5 When an integral is not analytically solvable (and then I suppose that I have to use Nintegrate), is it still possible to make the integral symbol appear in the code editor? In such a way to have more readable equations (especially when they are really long). I'm sorry for bothering you with this stupid beginner question, but I tried to read the documentation but I didn't manage to solve my issue. Thanks a lot in advance for your help.Here is R_1: ClearAll["Global*"]; n2 = 1; n1 = 1.38; Rs[\[Theta]_] = ( n1*Cos[\[Theta]] - n2*\[Sqrt](1 - (n1/n2*Sin[\[Theta]])^2))/( n1*Cos[\[Theta]] + n2*\[Sqrt](1 - (n1/n2*Sin[\[Theta]])^2)); Rp[\[Theta]_] = (-n2*Cos[\[Theta]] + n1*\[Sqrt](1 - (n1/n2*Sin[\[Theta]])^2))/( n2*Cos[\[Theta]] + n1*\[Sqrt](1 - (n1/n2*Sin[\[Theta]])^2)); R[\[Theta]_] = (Rs[\[Theta]] + Rp[\[Theta]])/2; c1 = \!$$\*SubsuperscriptBox[\(\[Integral]$$, $$0$$, $$\[Pi]/ 2$$]$$R[\[Theta]]*\ Sin[\[Theta]]* Cos[\[Theta]] \[DifferentialD]\[Theta]$$\) ; c2 = \!$$\*SubsuperscriptBox[\(\[Integral]$$, $$\(-\[Pi]$$/ 2\), $$0$$]$$R[\[Theta]]*\ Sin[\[Theta]]* \*SuperscriptBox[\((Cos[\[Theta]])$$, $$2$$] \[DifferentialD]\[Theta]\ \)\) ; R = (3 c2 + 2 c1)/(3 c2 - 2 c1 + 2) Here is R_2: enter code In[47]:= ClearAll["Global*"]; n1 = 1.38; n2 = 1; Rs[\[Theta]_] = Abs[((n1*Cos[\[Theta]] - n2*\[Sqrt](1 - (n1/n2*Sin[\[Theta]])^2))/( n1*Cos[\[Theta]] + n2*\[Sqrt](1 - (n1/n2*Sin[\[Theta]])^2)))]^2; In[51]:= Rp[\[Theta]_] = Abs[((n1*\[Sqrt](1 - (n1/n2*Sin[\[Theta]])^2) - n2*Cos[\[Theta]])/( n1*\[Sqrt](1 - (n1/n2*Sin[\[Theta]])^2) + n2*Cos[\[Theta]]))]^2; In[52]:= R[\[Theta]_] = (Rs[\[Theta]] + Rp[\[Theta]])/2; In[53]:= c1 = NIntegrate[ R[\[Theta]]* Sin[\[Theta]]*Cos[\[Theta]], {\[Theta], 0, \[Pi]/2}, MaxRecursion -> 100] During evaluation of In[53]:= NIntegrate::slwcon: Numerical integration converging too slowly; suspect one of the following: singularity, value of the integration is 0, highly oscillatory integrand, or WorkingPrecision too small. Out[53]= 0.256811 In[54]:= c2 = NIntegrate[ R[\[Theta]]* Sin[\[Theta]]*(Cos[\[Theta]])^2, {\[Theta], -(\[Pi]/2), 0}, MaxRecursion -> 100] During evaluation of In[54]:= NIntegrate::slwcon: Numerical integration converging too slowly; suspect one of the following: singularity, value of the integration is 0, highly oscillatory integrand, or WorkingPrecision too small. Out[54]= -0.12405 In[55]:= R = (3*c2 + 2*c1)/(3*c2 - 2*c1 + 2) Out[55]= 0.12697  Attachments:
 Gianluca Gorni 2 Votes The functions Rs and Rp become complex for \[Theta]0 = ArcSin[50/69]. You can insert that value into the integration interval as {\[Theta], 0, \[Theta]0, \[Pi]/2}, which silences the messages. You are using the symbol R with two different meanings, which may not be a good idea. The. I am afraid NIntegrate does not have a special notation.
 I am not an expert in integration. In your case for the integrals without Abs I could find an analytical primitive, but it was not easy to use, because it has a discontinuity in the imaginary part. Still, I think you can find an exact result in that case, with some patience. If you are content with a floating-point result, NIntegrate can be more than enough. I think that your warnings come because the algorithm somehow senses the singular point, but you have to study every case on its own.