**Triple Integral of sin(XYZ)dx dy dz**

To find the triple integral of sin(XYZ) with respect to x, y, and z, we will integrate the function over a region in three-dimensional space.

**Step 1: Determine the Integration Limits**
First, we need to determine the limits of integration for each variable. Let's assume we are integrating over a region R, which can be defined by the intervals [a, b], [c, d], and [e, f] for x, y, and z, respectively.

**Step 2: Set Up the Integral**
The triple integral of sin(XYZ) over region R can be expressed as:

∫∫∫ sin(XYZ) dxdydz

**Step 3: Integrate with Respect to x**
To integrate with respect to x, we treat y and z as constants. The integral becomes:

∫ sin(XYZ) dx

Integrating sin(XYZ) with respect to x gives us:

-1/XYZ * cos(XYZ) + C1,

where C1 is the constant of integration.

**Step 4: Integrate with Respect to y**
Next, we integrate the result obtained from the previous step with respect to y. This time, we treat z as a constant. The integral becomes:

∫ (-1/XYZ * cos(XYZ) + C1) dy

Integrating the expression above gives us:

-1/(XZ) * sin(XYZ) + C1y + C2,

where C1y is the constant of integration associated with the y variable, and C2 is the constant of integration.

**Step 5: Integrate with Respect to z**
Finally, we integrate the expression obtained from the previous step with respect to z. This time, all the variables are treated as constants. The integral becomes:

∫ (-1/(XZ) * sin(XYZ) + C1y + C2) dz

Integrating this expression gives us:

-1/XY * cos(XYZ) + C1yz + C2z + C3,

where C1yz is the constant of integration associated with the y and z variables, C2z is the constant of integration associated with the z variable, and C3 is the constant of integration.

**Step 6: Final Result**
After integrating with respect to all three variables, the triple integral of sin(XYZ) over the region R is:

-1/XY * cos(XYZ) + C1yz + C2z + C3,

where C1, C2, and C3 are the constants of integration obtained in the previous steps.

This is the final result of the triple integral of sin(XYZ)dx dy dz over the given region.