Let's break down the given integral to understand its structure and solve it step by step. The expression is:

$FY(y) = \int_0^1 \int_0^1 \int_0^{y} \frac{2e^{-5x_1^2}}{x_1x_2} e^{x_1 x_2 x_3} \, dx_3 \, dx_2 \, dx_1$

Step 1: Understanding the integral

• Outer integral (over $x_1$): This goes from 0 to 1.
• Middle integral (over $x_2$): This also goes from 0 to 1.
• Inner integral (over $x_3$): This goes from 0 to $y$, where $y$ is a parameter.

The integrand involves:

• $\frac{2e^{-5x_1^2}}{x_1 x_2}$, which depends on $x_1$ and $x_2$,
• $e^{x_1 x_2 x_3}$, which depends on all three variables $x_1, x_2, x_3$,
• Exponentials like $e^{-5x_1^2}$, which help to dampen the contribution of $x_1$.

Step 2: Solve the inner integral (over $x_3$)

The inner integral is:

$I_3 = \int_0^{y} e^{x_1 x_2 x_3} \, dx_3$

This is an exponential integral. We can compute it as:

$I_3 = \frac{1}{x_1 x_2} \left( e^{x_1 x_2 y} - 1 \right)$

Step 3: Substitute $I_3$ into the remaining integrals

Now, substitute the result of the $x_3$-integral into the remaining integrals. This gives:

$FY(y) = \int_0^1 \int_0^1 \frac{2e^{-5x_1^2}}{x_1 x_2} \left( \frac{e^{x_1 x_2 y} - 1}{x_1 x_2} \right) \, dx_2 \, dx_1$

This simplifies to:

$FY(y) = 2 \int_0^1 \int_0^1 \frac{e^{-5x_1^2}}{x_1^2 x_2^2} \left( e^{x_1 x_2 y} - 1 \right) \, dx_2 \, dx_1$

Step 4: Split the integral into two parts

The integrand has a subtraction, so we can split the integral into two parts:

$FY(y) = 2 \int_0^1 \int_0^1 \frac{e^{-5x_1^2}}{x_1^2 x_2^2} e^{x_1 x_2 y} \, dx_2 \, dx_1 - 2 \int_0^1 \int_0^1 \frac{e^{-5x_1^2}}{x_1^2 x_2^2} \, dx_2 \, dx_1$

Step 5: Simplify each part

First term:

The first term involves $e^{x_1 x_2 y}$, which can be handled as a standard exponential integral. To solve, we need a known method for integrating such terms, possibly using series expansion or special functions.

Second term:

The second term is simpler, as it has no dependence on $x_3$, so we would integrate it directly.

Conclusion

The full solution involves solving the two integrals, which may need further techniques such as series expansions or numerical methods for the non-trivial exponential integrals. Would you like me to proceed with approximating or numerically solving this, or is there a specific part you'd like to dive deeper into?