https://file.aiquickdraw.com/mathbot/file/1728143766596sln4htkl.jpg

The given expression is:

\[
2 \int_{0}^{\frac{\pi}{3}} \sqrt{2 + 2 \cos \theta} \, d\theta
\]

### Step 1: Simplifying the Square Root Expression
We can factor the term inside the square root as follows:

\[
\sqrt{2 + 2 \cos \theta} = \sqrt{2(1 + \cos \theta)} = \sqrt{2} \cdot \sqrt{1 + \cos \theta}
\]

Thus, the integral becomes:

\[
2 \int_{0}^{\frac{\pi}{3}} \sqrt{2} \cdot \sqrt{1 + \cos \theta} \, d\theta = 2\sqrt{2} \int_{0}^{\frac{\pi}{3}} \sqrt{1 + \cos \theta} \, d\theta
\]

### Step 2: Using a Trigonometric Identity
We use the identity:

\[
1 + \cos \theta = 2 \cos^2\left(\frac{\theta}{2}\right)
\]

So, the integral becomes:

\[
2\sqrt{2} \int_{0}^{\frac{\pi}{3}} \sqrt{2 \cos^2\left(\frac{\theta}{2}\right)} \, d\theta = 2\sqrt{2} \int_{0}^{\frac{\pi}{3}} \sqrt{2} \cos\left(\frac{\theta}{2}\right) \, d\theta
\]

This simplifies to:

\[
4 \int_{0}^{\frac{\pi}{3}} \cos\left(\frac{\theta}{2}\right) \, d\theta
\]

### Step 3: Changing the Variable of Integration
Let \( u = \frac{\theta}{2} \), so \( du = \frac{1}{2} d\theta \), and the limits change accordingly:
- When \( \theta = 0 \), \( u = 0 \).
- When \( \theta = \frac{\pi}{3} \), \( u = \frac{\pi}{6} \).

The integral becomes:

\[
4 \cdot 2 \int_{0}^{\frac{\pi}{6}} \cos u \, du = 8 \int_{0}^{\frac{\pi}{6}} \cos u \, du
\]

### Step 4: Integrating
The integral of \( \cos u \) is \( \sin u \). Thus, we have:

\[
8 [\sin u]_{0}^{\frac{\pi}{6}} = 8 \left( \sin\left(\frac{\pi}{6}\right) - \sin(0) \right) = 8 \left(\frac{1}{2}\right) = 4
\]

### Final Answer:
\[
\boxed{4}
\]

If you have any questions or would like more details, feel free to ask!

---

Here are 5 questions related to this problem:

1. What other trigonometric identities could be useful in similar integrals?
2. How can we compute definite integrals involving trigonometric functions more generally?
3. What are the key steps in handling integrals with square roots involving cosine or sine?
4. What role does variable substitution play in simplifying integrals?
5. Can this method be extended to other angles or limits of integration?

**Tip:** When dealing with integrals involving trigonometric functions, using appropriate identities like double-angle or half-angle formulas can greatly simplify the problem.

Evaluate the integral: \( 2 \int_{0}^{\frac{\pi}{3}} \sqrt{2 + 2 \cos \theta} \, d\theta \)

This problem demonstrates how to evaluate the integral \( 2 \int_{0}^{\frac{\pi}{3}} \sqrt{2 + 2 \cos \theta} \, d\theta \) using trigonometric identities and integration techniques. We simplify the expression inside the square root using \( 1 + \cos \theta = 2 \cos^2\left(\frac{\theta}{2}\right) \), change the variable of integration, and solve step-by-step. Suitable for students in Grades 11-12.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation