To evaluate the given integral by reversing the order of integration, we start by analyzing the limits of integration.

The original integral is:

$\int_1^{e^9} \int_{\ln x}^9 \frac{1}{e^y - y} \, dy \, dx$

Step 1: Identify the region of integration

• The outer integral for $x$ ranges from $1$ to $e^9$, i.e., $1 \leq x \leq e^9$.
• The inner integral for $y$ ranges from $\ln x$ to $9$, i.e., $\ln x \leq y \leq 9$.

We now switch the order of integration by expressing the limits of $x$ in terms of $y$.

Step 2: Find the new bounds

• For $x$, we have $x = e^y$, since $y = \ln x$.
• The lower bound for $x$ is $1$ and the upper bound is $e^9$, so:
  • When $y = 0$, $x = 1$.
  • When $y = 9$, $x = e^9$.

Thus, $y$ ranges from $0$ to $9$, and for a fixed $y$, $x$ ranges from $e^y$ to $e^9$.

Step 3: Rewrite the integral

The integral with the reversed order of integration becomes:

$\int_0^9 \int_{e^y}^{e^9} \frac{1}{e^y - y} \, dx \, dy$

Step 4: Evaluate the inner integral with respect to $x$

The inner integral is:

$\int_{e^y}^{e^9} \frac{1}{e^y - y} \, dx$

Since the integrand $\frac{1}{e^y - y}$ is independent of $x$, the integral simplifies to:

$\frac{1}{e^y - y} \int_{e^y}^{e^9} 1 \, dx = \frac{1}{e^y - y} (e^9 - e^y)$

Step 5: Simplify

This simplifies further to:

$(e^9 - e^y) \frac{1}{e^y - y}$

Step 6: Evaluate the outer integral

Now, we need to integrate this expression with respect to $y$:

$\int_0^9 (e^9 - e^y) \frac{1}{e^y - y} \, dy$

This is the final form of the integral after reversing the order of integration. However, solving this integral in a closed form may not be straightforward and could require advanced techniques or numerical methods.

Would you like to proceed with a numerical approximation or explore further details?

---

Related questions:

1. How can this integral be evaluated numerically?
2. What are the benefits of reversing the order of integration?
3. Can this problem be solved using substitution or special functions?
4. How would this change if the limits were different?
5. Can similar problems be solved using double integrals?

Tip: When reversing the order of integration, always sketch the region to ensure the limits are correct.