# Nested Integrate

I have the following equation

$$F(x,y,t) = \int_{0}^{\infty}\int_{-\infty}^{\infty}K(u,t)\cos[\epsilon(x-u)]\exp\left(-\sqrt{\epsilon^2 +y}\right)\text{d}u \text{d}\epsilon$$

How to express this function in mathematica ? Does NIntegrate[NIntegrate...] work?

• Please have a look at Menu/Help/WolframDocumentation/NIntegrate the second line from top. There you will find the answer. May 19, 2020 at 8:44

You can do it in one call, or split it into 2 calls to Integrate.

I prefer to split it so it is more clear for me. So you can do the inner integral, take the result, then do the outer integral.

ClearAll[u, t, k, e, x, y];
integrand = k[u, t] Cos[e*(x - u)]*Exp[-Sqrt[e^2 + y]]
int1 = Integrate[integrand, {u, -Infinity, Infinity}]


int2 = Integrate[int1, {e, 0, Infinity}]


But if you want to do it using one call then

ClearAll[u, t, k, e, x, y];
integrand = k[u, t] Cos[e*(x - u)]*Exp[-Sqrt[e^2 + y]]
Integrate[integrand, {e, 0, Infinity}, {u, -Infinity, Infinity}]