Let's analyze and solve the given double integral step by step.

The expression in the image is:

$\int_0^{\frac{\pi}{2}} \int_0^{\ln(2)} y \cos(x) e^{y \sin(x)} \, dy \, dx$

Step 1: Focus on the inner integral

We first solve the inner integral with respect to $y$, treating $x$ as a constant. The inner integral is:

$\int_0^{\ln(2)} y \cos(x) e^{y \sin(x)} \, dy$

The factor $\cos(x)$ is constant with respect to $y$, so it can be factored out:

$\cos(x) \int_0^{\ln(2)} y e^{y \sin(x)} \, dy$

Step 2: Solve the inner integral

We need to solve the integral:

$\int_0^{\ln(2)} y e^{y \sin(x)} \, dy$

This is an integration by parts problem. Let’s set:

• $u = y$, so $du = dy$,
• $dv = e^{y \sin(x)} \, dy$, so $v = \frac{1}{\sin(x)} e^{y \sin(x)}$ (using the chain rule).

Using integration by parts:

$\int u \, dv = uv - \int v \, du$

Substitute the values:

$y \cdot \frac{1}{\sin(x)} e^{y \sin(x)} \Bigg|_0^{\ln(2)} - \int_0^{\ln(2)} \frac{1}{\sin(x)} e^{y \sin(x)} \, dy$

Step 3: Simplify the result of integration by parts

Let’s compute the boundary terms:

At $y = \ln(2)$:

$\frac{\ln(2)}{\sin(x)} e^{\ln(2) \sin(x)} = \frac{\ln(2)}{\sin(x)} \cdot 2^{\sin(x)}$

At $y = 0$:

$0 \cdot e^{0} = 0$

Now the integral part:

$- \frac{1}{\sin(x)} \int_0^{\ln(2)} e^{y \sin(x)} \, dy = - \frac{1}{\sin(x)} \cdot \frac{1}{\sin(x)} e^{y \sin(x)} \Bigg|_0^{\ln(2)}$

Evaluating the limits:

At $y = \ln(2)$:

$\frac{1}{\sin(x)} \cdot e^{\ln(2) \sin(x)} = \frac{2^{\sin(x)}}{\sin(x)}$

At $y = 0$:

$\frac{1}{\sin(x)} \cdot e^0 = \frac{1}{\sin(x)}$

So the result of the inner integral becomes:

$\frac{\ln(2)}{\sin(x)} 2^{\sin(x)} - \left( \frac{2^{\sin(x)} - 1}{\sin^2(x)} \right)$

Step 4: Substitute the result into the outer integral

Now we substitute this result back into the outer integral:

$\int_0^{\frac{\pi}{2}} \cos(x) \left( \frac{\ln(2)}{\sin(x)} 2^{\sin(x)} - \frac{2^{\sin(x)} - 1}{\sin^2(x)} \right) dx$

This integral looks complicated, but it can be evaluated numerically if needed.

Step 5: Numerical evaluation

At this point, we can numerically compute the value of the outer integral using methods like numerical integration (e.g., Simpson's rule, trapezoidal rule, or software tools).

Would you like me to evaluate this integral numerically or explain more about the steps involved?

Here are some related questions:

1. What is the significance of integrating over different variables in this problem?
2. Can you show another example of applying integration by parts?
3. What happens if the limits of integration are changed?
4. How can we approximate such integrals using numerical methods?
5. How does the cosine factor affect the behavior of the integral?

Tip: When performing integration by parts, always carefully choose $u$ and $dv$ to simplify the resulting integrals.