\int _1^5\int _1^2\left(\frac{\ln \left(y\right)}{xy}\right)dydx

To solve the double integral:

\[
\int _1^5\int _1^2\left(\frac{\ln \left(y\right)}{xy}\right)dydx
\]

we will follow these steps:

### Step 1: Separate the Integrals
Notice that the integrand can be written as:

\[
\frac{\ln(y)}{xy} = \frac{1}{x} \cdot \frac{\ln(y)}{y}
\]

Thus, the integral becomes:

\[
\int_1^5 \frac{1}{x} \int_1^2 \frac{\ln(y)}{y} \, dy \, dx
\]

### Step 2: Evaluate the Inner Integral
First, compute the inner integral:

\[
\int_1^2 \frac{\ln(y)}{y} \, dy
\]

This integral is a standard one and can be solved using substitution:

\[
\int \frac{\ln(y)}{y} \, dy = \frac{1}{2}(\ln(y))^2 + C
\]

Applying the limits from 1 to 2:

\[
\int_1^2 \frac{\ln(y)}{y} \, dy = \frac{1}{2}(\ln(2))^2 - \frac{1}{2}(\ln(1))^2
\]

Since \(\ln(1) = 0\), this simplifies to:

\[
\int_1^2 \frac{\ln(y)}{y} \, dy = \frac{1}{2}(\ln(2))^2
\]

Let \( I = \frac{1}{2}(\ln(2))^2 \).

### Step 3: Evaluate the Outer Integral
Now, the integral becomes:

\[
I \int_1^5 \frac{1}{x} \, dx
\]

The integral \(\int_1^5 \frac{1}{x} \, dx\) is:

\[
\int_1^5 \frac{1}{x} \, dx = \ln(5) - \ln(1) = \ln(5)
\]

### Step 4: Multiply the Results
Now, multiply the results of the two integrals:

\[
\int_1^5\int_1^2 \frac{\ln(y)}{xy} \, dy \, dx = \frac{1}{2}(\ln(2))^2 \cdot \ln(5)
\]

Thus, the final answer is:

\[
\boxed{\frac{1}{2}(\ln(2))^2 \ln(5)}
\]

Would you like more details or have any questions?

### Related Questions:
1. How do you solve double integrals involving exponential functions?
2. What are the applications of double integrals in physics?
3. Can double integrals be solved using polar coordinates? When is it beneficial?
4. What substitution methods are useful in solving double integrals?
5. How do you interpret the result of a double integral in terms of area or volume?

### Tip:
Always look for ways to simplify the integrand before integrating, such as factoring out constants or using substitutions.

\int _1^5\int _1^2\left(\frac{\ln(y)}{xy}\right)dydx

This problem demonstrates how to solve a double integral involving logarithmic functions, focusing on integrating ln(y)/xy over the region 1 to 5 in x and 1 to 2 in y. The solution involves step-by-step integration techniques, using logarithmic identities and the Fundamental Theorem of Calculus.

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