The value of `int_(0)^(pi^(2) //4) ( sin sqrt(x))/( sqrt(x)) ` is

Options

A. `(pi)/( 4)`
B. `(pi^(2))/( 8)`
C. 2
D. 1

Answer

To solve the integral \( I = \int_{0}^{\frac{\pi^2}{4}} \frac{\sin(\sqrt{x})}{\sqrt{x}} \, dx \), we will use a substitution method.

### Step-by-Step Solution:

1. **Substitution**: Let \( t = \sqrt{x} \). Then, \( x = t^2 \) and \( dx = 2t \, dt \). The limits of integration change accordingly:
- When \( x = 0 \), \( t = \sqrt{0} = 0 \).
- When \( x = \frac{\pi^2}{4} \), \( t = \sqrt{\frac{\pi^2}{4}} = \frac{\pi}{2} \).

Thus, the integral becomes:
\[ I = \int_{0}^{\frac{\pi}{2}} \frac{\sin(t)}{t} (2t \, dt) = 2 \int_{0}^{\frac{\pi}{2}} \sin(t) \, dt \]

2. **Evaluate the Integral**: Now we need to evaluate \( \int_{0}^{\frac{\pi}{2}} \sin(t) \, dt \).
- The integral of \( \sin(t) \) is \( -\cos(t) \).
- Therefore,
\[ \int_{0}^{\frac{\pi}{2}} \sin(t) \, dt = \left[-\cos(t)\right]_{0}^{\frac{\pi}{2}} = -\cos\left(\frac{\pi}{2}\right) + \cos(0) = 0 + 1 = 1 \]

3. **Final Calculation**: Now substitute back into the expression for \( I \):
\[ I = 2 \cdot 1 = 2 \]

Thus, the value of the integral \( I \) is \( 2 \).