`int_0^(pi^2//4)("sin" sqrt(x))/(sqrtx) dx`

Options

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

Answer

To solve the integral \( \int_0^{\frac{\pi^2}{4}} \frac{\sin(\sqrt{x})}{\sqrt{x}} \, dx \), we will use the substitution method. Here’s a step-by-step solution:

### Step 1: Substitution
Let \( t = \sqrt{x} \). Then, we have:
\[ x = t^2 \quad \text{and} \quad dx = 2t \, dt \]

### Step 2: Change the limits of integration
When \( x = 0 \):
\[ t = \sqrt{0} = 0 \]
When \( x = \frac{\pi^2}{4} \):
\[ t = \sqrt{\frac{\pi^2}{4}} = \frac{\pi}{2} \]

### Step 3: Rewrite the integral
Now, substituting \( t \) into the integral, we get:
\[ \int_0^{\frac{\pi^2}{4}} \frac{\sin(\sqrt{x})}{\sqrt{x}} \, dx = \int_0^{\frac{\pi}{2}} \frac{\sin(t)}{t} (2t \, dt) \]
This simplifies to:
\[ = 2 \int_0^{\frac{\pi}{2}} \sin(t) \, dt \]

### Step 4: Integrate
The integral of \( \sin(t) \) is:
\[ \int \sin(t) \, dt = -\cos(t) \]
Thus,
\[ 2 \int_0^{\frac{\pi}{2}} \sin(t) \, dt = 2 \left[-\cos(t)\right]_0^{\frac{\pi}{2}} \]

### Step 5: Evaluate the limits
Now, we evaluate the limits:
\[ = 2 \left[-\cos\left(\frac{\pi}{2}\right) + \cos(0)\right] \]
Since \( \cos\left(\frac{\pi}{2}\right) = 0 \) and \( \cos(0) = 1 \), we have:
\[ = 2 \left[-0 + 1\right] = 2 \times 1 = 2 \]

### Final Answer
Thus, the value of the integral is:
\[ \boxed{2} \]