To evaluate the integral

$\int \log \left( \int \int \int y e^{x+y+z} \, dz \, dy \, dx \right),$

let’s proceed step by step:

Step 1: Understand the inner triple integral

The inner integral is:

$\int \int \int y e^{x+y+z} \, dz \, dy \, dx.$

Integrate with respect to $z$:

The term $y e^{x+y+z}$ depends on $z$ only through the exponential $e^{z}$. So, we treat $y e^{x+y}$ as a constant with respect to $z$:

$\int y e^{x+y+z} \, dz = y e^{x+y} \int e^z \, dz.$

The integral of $e^z$ is $e^z + C$. So, if the limits of integration for $z$ are $[z_1, z_2]$, then:

$\int_{z_1}^{z_2} y e^{x+y+z} \, dz = y e^{x+y} \left[ e^{z_2} - e^{z_1} \right].$

Integrate with respect to $y$:

Now, we consider:

$\int \left[ y e^{x+y} \left( e^{z_2} - e^{z_1} \right) \right] \, dy.$

Distribute $\left( e^{z_2} - e^{z_1} \right)$:

$\left( e^{z_2} - e^{z_1} \right) \int y e^{x+y} \, dy.$

Integrate with respect to $x$:

The final step is to integrate with respect to $x$. If limits for $x$ are not specified, the result will depend on these limits and yield an expression involving $x_1, x_2, y_1, y_2, z_1,$ and $z_2$.

Could you clarify the specific bounds of integration (for $x, y, z$) or confirm if they are over infinite or specific limits? This is crucial for completing the evaluation.

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Expansion Questions:

1. What are the specific bounds of integration for $x, y,$ and $z$?
2. Is the logarithm applied to the entire triple integral, or is it a part of the integrand?
3. Are there specific simplifications or transformations you wish to apply?
4. Should this be evaluated symbolically or numerically with specific bounds?
5. What is the context or application of this integral?

Tip:

When evaluating triple integrals, carefully separate the dependency of each variable in the exponential term to simplify computations effectively.