If `y=int_(0)^(x)sqrt(sin x)dx` the value of `(dy)/(dx)` at `x=(pi)/(2)` is :

Options

A. 0
B. 1
C. `-1`
D. None of these

Answer

To solve the problem, we need to find the derivative of the integral \( y = \int_{0}^{x} \sqrt{\sin t} \, dt \) at the point \( x = \frac{\pi}{2} \).

### Step-by-Step Solution:

1. **Define the integral**:
\[ y = \int_{0}^{x} \sqrt{\sin t} \, dt \]

2. **Differentiate using Leibniz's rule**:
According to Leibniz's rule for differentiation under the integral sign, if \( y = \int_{a}^{g(x)} f(t) \, dt \), then:
\[ \frac{dy}{dx} = f(g(x)) \cdot g'(x) \]
Here, \( f(t) = \sqrt{\sin t} \), \( g(x) = x \), and \( a = 0 \).

3. **Calculate \( g'(x) \)**:
Since \( g(x) = x \), we have:
\[ g'(x) = 1 \]

4. **Apply Leibniz's rule**:
Therefore, we can write:
\[ \frac{dy}{dx} = \sqrt{\sin x} \cdot 1 = \sqrt{\sin x} \]

5. **Evaluate at \( x = \frac{\pi}{2} \)**:
Now we need to find \( \frac{dy}{dx} \) at \( x = \frac{\pi}{2} \):
\[ \frac{dy}{dx} \bigg|_{x = \frac{\pi}{2}} = \sqrt{\sin\left(\frac{\pi}{2}\right)} \]

6. **Calculate \( \sin\left(\frac{\pi}{2}\right) \)**:
We know that:
\[ \sin\left(\frac{\pi}{2}\right) = 1 \]

7. **Final calculation**:
Therefore:
\[ \frac{dy}{dx} \bigg|_{x = \frac{\pi}{2}} = \sqrt{1} = 1 \]

### Conclusion:
The value of \( \frac{dy}{dx} \) at \( x = \frac{\pi}{2} \) is \( 1 \).

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