not that, evaluate*M*=∫0 to 5*π/6* ​​(1+sin(*x*))*dx*

To evaluate the integral \( M = \int_{0}^{\frac{5\pi}{6}} (1 + \sin(x)) \, dx \), we need to compute the integral of the function \( 1 + \sin(x) \) over the interval from \( 0 \) to \( \frac{5\pi}{6} \).

### Step-by-step Solution:

1. **Separate the Integral:**

   The given integral can be separated into two integrals:
   \[
   M = \int_{0}^{\frac{5\pi}{6}} 1 \, dx + \int_{0}^{\frac{5\pi}{6}} \sin(x) \, dx
   \]

2. **Evaluate Each Integral:**

   - For the first integral:
     \[
     \int_{0}^{\frac{5\pi}{6}} 1 \, dx = \left[ x \right]_{0}^{\frac{5\pi}{6}} = \frac{5\pi}{6} - 0 = \frac{5\pi}{6}
     \]

   - For the second integral:
     \[
     \int_{0}^{\frac{5\pi}{6}} \sin(x) \, dx = \left[ -\cos(x) \right]_{0}^{\frac{5\pi}{6}}
     \]
     Evaluate at the limits:
     \[
     -\cos\left(\frac{5\pi}{6}\right) - (-\cos(0)) = -\left(-\frac{\sqrt{3}}{2}\right) - (-1) = \frac{\sqrt{3}}{2} + 1
     \]

3. **Combine the Results:**

   Combining the two integrals, we get:
   \[
   M = \frac{5\pi}{6} + \left( \frac{\sqrt{3}}{2} + 1 \right) = \frac{5\pi}{6} + \frac{\sqrt{3}}{2} + 1
   \]

### Final Answer:

\[
\boxed{M = \frac{5\pi}{6} + \frac{\sqrt{3}}{2} + 1}
\]

Would you like more details on any of the steps, or have any questions?

### Related Questions:

1. How do you evaluate integrals involving trigonometric functions?
2. What is the integral of \(\sin(x)\) and \(\cos(x)\)?
3. How does the Fundamental Theorem of Calculus help in evaluating definite integrals?
4. How do you compute the area under a curve for trigonometric functions?
5. How can integrals be used to calculate the average value of a function over an interval?

### Tip:

When evaluating integrals, breaking them down into simpler parts often makes the process easier and more manageable.

Evaluate M = ∫0 to 5π/6 (1 + sin(x)) dx

This page provides a step-by-step solution to the integral ∫0 to 5π/6 (1 + sin(x)) dx. It covers key calculus concepts, including splitting the integral into simpler parts and applying trigonometric integrals. Suitable for first-year undergraduate calculus students.

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