Choose the correct Answer of the Following Questions : `int_0^(1/2) -1/sqrt(1-x^2)dx =`

Options

A. `(pi/6)`
B. `(pi/3)`
C. `(pi/2)`
D. `(3pi/4)`

Answer

To solve the definite integral \( \int_0^{1/2} -\frac{1}{\sqrt{1-x^2}} \, dx \), we can follow these steps:

### Step 1: Identify the Integral
We start with the integral:
\[ I = \int_0^{1/2} -\frac{1}{\sqrt{1-x^2}} \, dx \]

### Step 2: Recognize the Integral's Antiderivative
The function \( -\frac{1}{\sqrt{1-x^2}} \) is the derivative of \( \cos^{-1}(x) \). Therefore, we can rewrite the integral as:
\[ I = -\int_0^{1/2} \frac{1}{\sqrt{1-x^2}} \, dx = -\left[ \cos^{-1}(x) \right]_0^{1/2} \]

### Step 3: Evaluate the Antiderivative at the Limits
Now we evaluate \( \cos^{-1}(x) \) at the limits \( 0 \) and \( \frac{1}{2} \):
\[ I = -\left( \cos^{-1}\left(\frac{1}{2}\right) - \cos^{-1}(0) \right) \]

### Step 4: Calculate \( \cos^{-1}\left(\frac{1}{2}\right) \) and \( \cos^{-1}(0) \)
We know:
- \( \cos^{-1}\left(\frac{1}{2}\right) = \frac{\pi}{3} \)
- \( \cos^{-1}(0) = \frac{\pi}{2} \)

### Step 5: Substitute the Values
Substituting these values into the integral gives:
\[ I = -\left( \frac{\pi}{3} - \frac{\pi}{2} \right) \]

### Step 6: Simplify the Expression
Now, we simplify the expression:
\[ I = -\left( \frac{\pi}{3} - \frac{3\pi}{6} \right) = -\left( \frac{\pi}{3} - \frac{2\pi}{6} \right) = -\left( \frac{2\pi}{6} - \frac{3\pi}{6} \right) = -\left( -\frac{\pi}{6} \right) = \frac{\pi}{6} \]

### Step 7: Final Answer
Thus, the value of the definite integral is:
\[ \int_0^{1/2} -\frac{1}{\sqrt{1-x^2}} \, dx = \frac{\pi}{6} \]