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To evaluate the integral 

\[
\int_0^{\frac{\pi}{2}} \cos^2(x) \, dx
\]

we can simplify it by using a trigonometric identity.

### Step 1: Use the Power-Reducing Identity
The identity for \(\cos^2(x)\) is:
\[
\cos^2(x) = \frac{1 + \cos(2x)}{2}.
\]

### Step 2: Substitute in the Integral
Substituting \(\cos^2(x)\) with \(\frac{1 + \cos(2x)}{2}\), we get:
\[
\int_0^{\frac{\pi}{2}} \cos^2(x) \, dx = \int_0^{\frac{\pi}{2}} \frac{1 + \cos(2x)}{2} \, dx.
\]

### Step 3: Separate the Integral
This can be split into two integrals:
\[
\int_0^{\frac{\pi}{2}} \cos^2(x) \, dx = \int_0^{\frac{\pi}{2}} \frac{1}{2} \, dx + \int_0^{\frac{\pi}{2}} \frac{\cos(2x)}{2} \, dx.
\]

### Step 4: Evaluate Each Integral Separately
1. **First Integral:**
   \[
   \int_0^{\frac{\pi}{2}} \frac{1}{2} \, dx = \frac{1}{2} \int_0^{\frac{\pi}{2}} 1 \, dx = \frac{1}{2} \cdot \frac{\pi}{2} = \frac{\pi}{4}.
   \]

2. **Second Integral:**
   \[
   \int_0^{\frac{\pi}{2}} \frac{\cos(2x)}{2} \, dx = \frac{1}{2} \int_0^{\frac{\pi}{2}} \cos(2x) \, dx.
   \]
   To evaluate \(\int \cos(2x) \, dx\), we use the antiderivative:
   \[
   \int \cos(2x) \, dx = \frac{\sin(2x)}{2}.
   \]
   Applying the limits from \(0\) to \(\frac{\pi}{2}\):
   \[
   \frac{1}{2} \cdot \left[ \frac{\sin(2x)}{2} \right]_0^{\frac{\pi}{2}} = \frac{1}{2} \cdot \left( \frac{\sin(\pi)}{2} - \frac{\sin(0)}{2} \right) = \frac{1}{2} \cdot (0 - 0) = 0.
   \]

### Step 5: Combine Results
Thus,
\[
\int_0^{\frac{\pi}{2}} \cos^2(x) \, dx = \frac{\pi}{4} + 0 = \frac{\pi}{4}.
\]

### Final Answer
\[
\int_0^{\frac{\pi}{2}} \cos^2(x) \, dx = \frac{\pi}{4}.
\]

Would you like more details or have any questions?

---

1. How would the result change if the limits of integration were different?
2. How can you apply this power-reducing identity to \(\sin^2(x)\)?
3. What if the integrand were \(\sin^2(x) + \cos^2(x)\)?
4. How would you evaluate \(\int \cos^2(x) \sin(x) \, dx\)?
5. Can you apply this method to find \(\int_0^{2\pi} \cos^2(x) \, dx\)?

**Tip:** The power-reducing identities are particularly useful for integrals involving \(\sin^2(x)\) or \(\cos^2(x)\) over standard intervals, as they simplify the expression into terms that are easier to integrate.

Evaluate the integral \( \int_0^{\frac{\pi}{2}} \cos^2(x) \, dx \)

Learn how to evaluate the integral \( \int_0^{\frac{\pi}{2}} \cos^2(x) \, dx \) using trigonometric identities. This solution covers the power-reducing identity and a step-by-step approach to integrate the function, ideal for college calculus students.

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