`int_(0) ^(pi//2) cos^(2) theta d theta =`

Options

A. 0
B. 2
C. `pi/4`
D. `pi/2`

Answer

To solve the integral \( \int_{0}^{\frac{\pi}{2}} \cos^2 \theta \, d\theta \), we can use a clever technique involving symmetry and the properties of trigonometric functions. Here’s a step-by-step breakdown of the solution:

### Step 1: Define the Integral
Let
\[ I = \int_{0}^{\frac{\pi}{2}} \cos^2 \theta \, d\theta \]

### Step 2: Use the Identity for Cosine
We can use the identity \( \cos^2 \theta = \cos^2\left(\frac{\pi}{2} - \theta\right) \). Therefore, we can rewrite the integral as:
\[ I = \int_{0}^{\frac{\pi}{2}} \cos^2\left(\frac{\pi}{2} - \theta\right) \, d\theta \]
Since \( \cos\left(\frac{\pi}{2} - \theta\right) = \sin \theta \), we have:
\[ I = \int_{0}^{\frac{\pi}{2}} \sin^2 \theta \, d\theta \]

### Step 3: Add the Two Integrals
Now, we can add the two expressions for \( I \):
\[ 2I = \int_{0}^{\frac{\pi}{2}} \cos^2 \theta \, d\theta + \int_{0}^{\frac{\pi}{2}} \sin^2 \theta \, d\theta \]
This simplifies to:
\[ 2I = \int_{0}^{\frac{\pi}{2}} \left(\cos^2 \theta + \sin^2 \theta\right) \, d\theta \]

### Step 4: Use the Pythagorean Identity
Using the Pythagorean identity \( \cos^2 \theta + \sin^2 \theta = 1 \), we can simplify the integral:
\[ 2I = \int_{0}^{\frac{\pi}{2}} 1 \, d\theta \]

### Step 5: Evaluate the Integral
Now, we can evaluate the integral:
\[ 2I = \left[ \theta \right]_{0}^{\frac{\pi}{2}} = \frac{\pi}{2} - 0 = \frac{\pi}{2} \]

### Step 6: Solve for \( I \)
Now, we can solve for \( I \):
\[ I = \frac{\pi}{4} \]

### Final Answer
Thus, the value of the integral is:
\[ \int_{0}^{\frac{\pi}{2}} \cos^2 \theta \, d\theta = \frac{\pi}{4} \]

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