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**Problem:**
\[ \text{Evaluate } \int \sin^2 \left(\frac{\pi x}{7}\right) \, dx \text{ using the half-angle identity.} \]

**Solution:**

1. **Use the Half-Angle Identity**:
   The half-angle identity for sine is given by:
   \[
   \sin^2 \theta = \frac{1 - \cos(2\theta)}{2}
   \]
   In this problem, let \(\theta = \frac{\pi x}{7}\). Then:
   \[
   \sin^2 \left(\frac{\pi x}{7}\right) = \frac{1 - \cos\left(\frac{2\pi x}{7}\right)}{2}
   \]

2. **Rewrite the Integral**:
   Substitute the identity into the integral:
   \[
   \int \sin^2 \left(\frac{\pi x}{7}\right) \, dx = \int \frac{1 - \cos\left(\frac{2\pi x}{7}\right)}{2} \, dx
   \]

   This simplifies to:
   \[
   \frac{1}{2} \int \left(1 - \cos\left(\frac{2\pi x}{7}\right)\right) dx
   \]

3. **Integrate**:
   Now, integrate each term separately:
   \[
   \frac{1}{2} \left( \int 1 \, dx - \int \cos\left(\frac{2\pi x}{7}\right) \, dx \right)
   \]

   - The integral of \(1\) with respect to \(x\) is \(x\).
   - The integral of \(\cos\left(\frac{2\pi x}{7}\right)\) can be computed using the substitution method.

   Let \( u = \frac{2\pi x}{7} \), so \( du = \frac{2\pi}{7} dx \) or \( dx = \frac{7}{2\pi} du \).

   The integral becomes:
   \[
   \int \cos\left(\frac{2\pi x}{7}\right) dx = \frac{7}{2\pi} \int \cos(u) du = \frac{7}{2\pi} \sin(u)
   \]

   Substituting back for \( u \):
   \[
   \frac{7}{2\pi} \sin\left(\frac{2\pi x}{7}\right)
   \]

4. **Final Answer**:
   Combine the results to get the final solution:
   \[
   \frac{1}{2} \left( x - \frac{7}{2\pi} \sin\left(\frac{2\pi x}{7}\right) \right) + C
   \]
   Simplifying further:
   \[
   \boxed{\frac{x}{2} - \frac{7}{4\pi} \sin\left(\frac{2\pi x}{7}\right) + C}
   \]
   where \( C \) is the constant of integration.

Would you like any further details or explanations on this problem? 

Here are 5 related questions you might find useful:
1. How would you solve a similar integral involving \(\cos^2 \left(\frac{\pi x}{7}\right)\)?
2. What other trigonometric identities could be used to solve integrals like this?
3. How does the substitution method work in solving integrals?
4. Can you apply the same steps to evaluate \(\int \sin^2 \left(\frac{2\pi x}{7}\right) \, dx\)?
5. What are some common mistakes to avoid when applying the half-angle identity?

**Tip:** When solving integrals involving trigonometric functions, always consider simplifying the expression using identities like the half-angle or double-angle formulas before integrating.

Evaluate ∫ sin^2(πx/7) dx using the half-angle identity.

Step-by-step solution for evaluating the integral of sin^2(πx/7) using the half-angle identity. Learn how to rewrite the trigonometric function and solve the integral with detailed explanations, suitable for university-level calculus students.

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