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The given problem is to evaluate the double integral:

\[
\int_0^{\frac{\pi}{4}} \int_2^7 (y \cos x + 1) \, dy \, dx.
\]

Let's break down and solve this step by step.

### Step 1: Integrate with respect to \(y\) (the inner integral)
The inner integral is:
\[
\int_2^7 (y \cos x + 1) \, dy.
\]
We can split this integral as:
\[
\int_2^7 y \cos x \, dy + \int_2^7 1 \, dy.
\]

#### First term:
\[
\int_2^7 y \cos x \, dy = \cos x \int_2^7 y \, dy.
\]
Now, integrate \(y\):
\[
\int_2^7 y \, dy = \left[\frac{y^2}{2}\right]_2^7 = \frac{7^2}{2} - \frac{2^2}{2} = \frac{49}{2} - \frac{4}{2} = \frac{45}{2}.
\]
So,
\[
\int_2^7 y \cos x \, dy = \cos x \cdot \frac{45}{2} = \frac{45}{2} \cos x.
\]

#### Second term:
\[
\int_2^7 1 \, dy = \left[y\right]_2^7 = 7 - 2 = 5.
\]

Thus, the inner integral becomes:
\[
\int_2^7 (y \cos x + 1) \, dy = \frac{45}{2} \cos x + 5.
\]

### Step 2: Integrate with respect to \(x\) (the outer integral)
Now, we need to evaluate:
\[
\int_0^{\frac{\pi}{4}} \left(\frac{45}{2} \cos x + 5\right) \, dx.
\]
This can be split as:
\[
\frac{45}{2} \int_0^{\frac{\pi}{4}} \cos x \, dx + 5 \int_0^{\frac{\pi}{4}} 1 \, dx.
\]

#### First term:
\[
\int_0^{\frac{\pi}{4}} \cos x \, dx = \sin x \bigg|_0^{\frac{\pi}{4}} = \sin\left(\frac{\pi}{4}\right) - \sin(0) = \frac{\sqrt{2}}{2} - 0 = \frac{\sqrt{2}}{2}.
\]
So,
\[
\frac{45}{2} \int_0^{\frac{\pi}{4}} \cos x \, dx = \frac{45}{2} \cdot \frac{\sqrt{2}}{2} = \frac{45 \sqrt{2}}{4}.
\]

#### Second term:
\[
\int_0^{\frac{\pi}{4}} 1 \, dx = \left[x\right]_0^{\frac{\pi}{4}} = \frac{\pi}{4} - 0 = \frac{\pi}{4}.
\]
So,
\[
5 \int_0^{\frac{\pi}{4}} 1 \, dx = 5 \cdot \frac{\pi}{4} = \frac{5\pi}{4}.
\]

### Step 3: Combine the results
Now, combine the two results:
\[
\frac{45 \sqrt{2}}{4} + \frac{5\pi}{4}.
\]

Thus, the value of the double integral is:
\[
\boxed{\frac{45 \sqrt{2}}{4} + \frac{5\pi}{4}}.
\]

Let me know if you'd like further details on any of these steps!

### Related Questions:
1. How do you handle switching the order of integration in double integrals?
2. What happens when the limits of integration depend on each other?
3. Can you evaluate this integral using polar coordinates?
4. How does the cosine function affect the symmetry of the integral?
5. What are some common strategies for simplifying integrals like this one?

### Tip:
When solving double integrals, it often helps to break them down step-by-step, starting with the inner integral and working outward.

Evaluate the integral \( \int_0^{\frac{\pi}{4}} \int_2^7 (y \cos x + 1) \, dy \, dx. \)

This page demonstrates how to evaluate the double integral \( \int_0^{\frac{\pi}{4}} \int_2^7 (y \cos x + 1) \, dy \, dx \), including step-by-step solutions that apply calculus concepts such as integration of polynomials and trigonometric functions. Suitable for Calculus students.

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