https://r2.erweima.ai/mathbot/image/1725495080486zcrmt514.jpg

The image contains an integral problem where we are tasked with calculating the value of \( M \). Let's break down the problem in parts.

### Problem Summary:
1. **Given Integral:**
   \[
   I = \int \left[ \cos(2x) \sqrt{1 + 2\sin(2x)} + \frac{5e^{2x}}{3 + 2e^{2x}} \right] dx
   \]
   
2. **Solution Form:**
   The result is given in the form:
   \[
   I = d(p + 2\sin(nx))^m + a \ln|b + 2e^{cx}| + K
   \]
   where \( d, p, n, m, a, b, c \) are constants.

3. **Find the Value of \( M \):**
   The expression for \( M \) is:
   \[
   M = \frac{a \cdot b \cdot c \cdot d \cdot m \cdot n}{2p}
   \]
   
4. **Hint:**
   A suggested substitution is given: \( u = 1 + 2 \sin(2x) \) for the first term of the integral.

### Solution Approach:

We need to:
- Solve the integral using the suggested substitution for the first term.
- Match the resulting expression with the given solution form to identify the constants \( a, b, c, d, p, n, m \).
- Finally, use these constants to compute the value of \( M \).

Would you like to proceed with solving the integral step by step, applying the substitution?

Calculate the integral: I = ∫ [cos(2x)√(1 + 2sen(2x)) + (5e^(2x))/(3 + 2e^(2x))] dx, with the result given as I = d(p + 2sen(nx))^m + a ln|b + 2e^(cx)| + K. Find the value of M = (a * b * c * d * m * n) / 2p. Hint: Use the variable substitution u = 1 + 2sen(2x) for the first term.

This problem focuses on solving a complex integral involving trigonometric and exponential functions using the substitution method. The solution leads to finding the constants a, b, c, d, m, n, and computing the value of M.

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