Consider `A =int_(0)^((pi)/(4))(sin(2x))/(x)dx,` then

Options

A. `A gt (pi)/(2)`
B. `A =(pi)/(2)`
C. `A lt (pi)/(2)`
D. `A gt pi`

Answer

To solve the integral \( A = \int_{0}^{\frac{\pi}{4}} \frac{\sin(2x)}{x} \, dx \), we will analyze the behavior of the function involved and apply some properties of integrals.

### Step-by-Step Solution:

1. **Understanding the Function**:
We know that for \( x \) in the interval \( (0, \frac{\pi}{4}) \), the function \( \sin(2x) \) is less than \( 2x \). This is derived from the fact that \( \sin \theta < \theta \) for \( \theta \) in \( (0, \frac{\pi}{2}) \).

**Hint**: Recall that \( \sin \theta < \theta \) for \( \theta \) in \( (0, \frac{\pi}{2}) \).

2. **Setting Up the Inequality**:
From the above, we can write:
\[ \frac{\sin(2x)}{2x} < 1 \]
This implies:
\[ \sin(2x) < 2x \]

3. **Rewriting the Integral**:
We can multiply and divide the integrand by 2:
\[ A = \int_{0}^{\frac{\pi}{4}} \frac{\sin(2x)}{x} \, dx = \int_{0}^{\frac{\pi}{4}} \frac{\sin(2x)}{2x} \cdot 2 \, dx \]

4. **Using the Inequality**:
Since \( \frac{\sin(2x)}{2x} < 1 \), we have:
\[ A < \int_{0}^{\frac{\pi}{4}} 1 \, dx \]

5. **Calculating the Integral**:
The integral of 1 from 0 to \( \frac{\pi}{4} \) is:
\[ \int_{0}^{\frac{\pi}{4}} 1 \, dx = \left[ x \right]_{0}^{\frac{\pi}{4}} = \frac{\pi}{4} - 0 = \frac{\pi}{4} \]

6. **Final Inequality**:
Therefore, we can conclude:
\[ A < \frac{\pi}{4} \cdot 2 = \frac{\pi}{2} \]

### Conclusion:
Thus, we have shown that:
\[ A < \frac{\pi}{2} \]

The correct option is **C**.