Final answer:
To evaluate the iterated integral ∫∫∫ e² / (xz ln(z)) dy dz dx, where 0 ≤ x ≤ 3, 1 ≤ y ≤ 2, 1 ≤ z ≤ e, follow the order of integration and integrate step by step with respect to y, z, and x.
Step-by-step explanation:
To evaluate the iterated integral ∫∫∫ e² / (xz ln(z)) dy dz dx, where 0 ≤ x ≤ 3, 1 ≤ y ≤ 2, 1 ≤ z ≤ e, we can simply follow the order of integration. Let's start with the innermost integral:
∫ e² / (xz ln(z)) dy dz dx = ∫1e ∫12 ∫03 e² / (xz ln(z)) dy dz dx
Integrating with respect to y, the integral becomes:
∫ e² / (xz ln(z)) dy dz dx = ∫1e ∫12 e²y / (xz ln(z)) dz dx
Next, integrate with respect to z:
∫ e² / (xz ln(z)) dy dz dx = ∫1e e² * [ln(z)]12 / (xz) dx
Finally, integrate with respect to x:
∫ e² / (xz ln(z)) dy dz dx = ∫1e e² * [ln(z)]12 * ln(x)