# Difficult definite integral

I want to compute a numerical value for the following definite integral: $$\int_0^1\int_0^1 e^{-\xi^2+\eta^2}e^{\int_0^{1/2}e^{-2\,\xi\,(1+\cos(3 s))}\eta\sin (\pi s)\,ds}\,d\xi\,d\eta.$$ I wrote in Mathematica the following code:

Integrate[Exp[-xi^2 + eta^2]*Exp[Integrate[Exp[-2*xi*(1+Cos[3*s])]*eta*Sin[Pi*s], {s,0, 0.5}]], {xi, 0, 1}, {eta, 0, 1}]


I think the problem is the computation of $$\int_0^{1/2}e^{-2\,\xi\,(1+\cos(3 s))}\eta\sin (\pi s)\,ds.$$ Is there a way to compute integrals of the form $$\int_a^b F(x,y)\,dy$$ in Mathematica? Integrals of the form $\int_a^b F(y)\,dy$ can be computed exactly or numerically, with the Integral or NIntegral functions, but what about $\int_a^b F(x,y)\,dy$ ?

• Does this solve your problem? – Lukas Lang Sep 16 '17 at 12:23
• @Mathe172 Yes! Thank you very much. – user39756 Sep 16 '17 at 12:49

i1[xi_?NumericQ, eta_?NumericQ] := i1[xi, eta] = NIntegrate[Exp[-2*xi*(1 + Cos[3*s])]*eta*Sin[Pi*s], {s, 0, 0.5}];