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.