If `I=int_(0)^(1//2) (1)/(sqrt(1-x^(2n)))dx`then which one of the following is not true ?

Options

A. `I lt (pi)/(6)`
B. `I ge (1)/(2)`
C. `I gt 0`
D. none of these

Answer

To solve the problem, we need to analyze the integral \( I = \int_{0}^{\frac{1}{2}} \frac{1}{\sqrt{1 - x^{2n}}} \, dx \) and determine which of the given statements about \( I \) is not true.

### Step 1: Set up the integral
We have:
\[ I = \int_{0}^{\frac{1}{2}} \frac{1}{\sqrt{1 - x^{2n}}} \, dx \]

### Step 2: Analyze the function under the integral
For \( x \) in the interval \( [0, \frac{1}{2}] \) and \( n > 1 \), we can observe that \( x^{2n} < x^2 \). Thus, we can state:
\[ 1 - x^{2n} > 1 - x^2 \]
Taking the square root:
\[ \sqrt{1 - x^{2n}} > \sqrt{1 - x^2} \]

### Step 3: Invert the inequality
This implies:
\[ \frac{1}{\sqrt{1 - x^{2n}}} < \frac{1}{\sqrt{1 - x^2}} \]

### Step 4: Integrate both sides
Now we can integrate both sides from \( 0 \) to \( \frac{1}{2} \):
\[ \int_{0}^{\frac{1}{2}} \frac{1}{\sqrt{1 - x^{2n}}} \, dx < \int_{0}^{\frac{1}{2}} \frac{1}{\sqrt{1 - x^2}} \, dx \]
Thus:
\[ I < \int_{0}^{\frac{1}{2}} \frac{1}{\sqrt{1 - x^2}} \, dx \]

### Step 5: Evaluate the right-hand side integral
The integral \( \int_{0}^{\frac{1}{2}} \frac{1}{\sqrt{1 - x^2}} \, dx \) can be evaluated as:
\[ \int \frac{1}{\sqrt{1 - x^2}} \, dx = \sin^{-1}(x) + C \]
Evaluating from \( 0 \) to \( \frac{1}{2} \):
\[ \sin^{-1}\left(\frac{1}{2}\right) - \sin^{-1}(0) = \frac{\pi}{6} - 0 = \frac{\pi}{6} \]

### Step 6: Conclude the comparison
Thus, we have:
\[ I < \frac{\pi}{6} \]
Now, since \( \frac{\pi}{6} \approx 0.523 \), we can conclude:
\[ I < \frac{5}{6} \quad \text{(since } \frac{5}{6} \approx 0.833 \text{)} \]

### Step 7: Check the other inequalities
We also know:
1. \( I > 0 \) (since the integrand is positive).
2. We need to check if \( I \geq \frac{1}{2} \). Given that \( I \) is less than \( \frac{5}{6} \) but greater than \( 0 \), we can conclude that \( I \) could be less than \( \frac{1}{2} \) depending on the value of \( n \).

### Conclusion
Thus, we can summarize:
- \( I < \frac{5}{6} \) is true.
- \( I > 0 \) is true.
- \( I \geq \frac{1}{2} \) may not be true for all \( n > 1 \).

Therefore, the statement that is **not true** is:
- \( I \geq \frac{1}{2} \).