To solve the integral \( \int_0^{1/2} \frac{x^2}{(1-x^2)^{3/2}} \, dx \) and find \( k \) such that \( \int_0^{1/2} \frac{x^2}{(1-x^2)^{3/2}} \, dx = \frac{k}{6} \), we can follow these steps:
### Step 1: Substitution
Let \( x = \sin \theta \). Then, \( dx = \cos \theta \, d\theta \).
### Step 2: Change the limits of integration
When \( x = 0 \), \( \theta = 0 \).
When \( x = \frac{1}{2} \), \( \theta = \sin^{-1}(\frac{1}{2}) = \frac{\pi}{6} \).
### Step 3: Rewrite the integral
The integral becomes:
\[
\int_0^{\frac{\pi}{6}} \frac{\sin^2 \theta}{(1 - \sin^2 \theta)^{3/2}} \cos \theta \, d\theta
\]
Since \( 1 - \sin^2 \theta = \cos^2 \theta \), we can rewrite the integral as:
\[
\int_0^{\frac{\pi}{6}} \frac{\sin^2 \theta}{\cos^3 \theta} \cos \theta \, d\theta = \int_0^{\frac{\pi}{6}} \frac{\sin^2 \theta}{\cos^2 \theta} \, d\theta
\]
This simplifies to:
\[
\int_0^{\frac{\pi}{6}} \tan^2 \theta \, d\theta
\]
### Step 4: Use the identity for \( \tan^2 \theta \)
Recall that \( \tan^2 \theta = \sec^2 \theta - 1 \). Thus, we can rewrite the integral:
\[
\int_0^{\frac{\pi}{6}} \tan^2 \theta \, d\theta = \int_0^{\frac{\pi}{6}} (\sec^2 \theta - 1) \, d\theta
\]
### Step 5: Evaluate the integral
Now, we can split the integral:
\[
\int_0^{\frac{\pi}{6}} \sec^2 \theta \, d\theta - \int_0^{\frac{\pi}{6}} 1 \, d\theta
\]
The integral of \( \sec^2 \theta \) is \( \tan \theta \), so we have:
\[
\left[ \tan \theta \right]_0^{\frac{\pi}{6}} - \left[ \theta \right]_0^{\frac{\pi}{6}} = \tan\left(\frac{\pi}{6}\right) - 0 - \left(\frac{\pi}{6} - 0\right)
\]
Calculating \( \tan\left(\frac{\pi}{6}\right) = \frac{1}{\sqrt{3}} \):
\[
\frac{1}{\sqrt{3}} - \frac{\pi}{6}
\]
### Step 6: Final result
Thus, we have:
\[
\int_0^{1/2} \frac{x^2}{(1-x^2)^{3/2}} \, dx = \frac{1}{\sqrt{3}} - \frac{\pi}{6}
\]
Setting this equal to \( \frac{k}{6} \):
\[
\frac{1}{\sqrt{3}} - \frac{\pi}{6} = \frac{k}{6}
\]
Multiplying through by 6 gives:
\[
6\left(\frac{1}{\sqrt{3}} - \frac{\pi}{6}\right) = k
\]
This simplifies to:
\[
k = 6 \cdot \frac{1}{\sqrt{3}} - \pi = \frac{6}{\sqrt{3}} - \pi = 2\sqrt{3} - \pi
\]
### Final Answer
Thus, the value of \( k \) is:
\[
\boxed{2\sqrt{3} - \pi}
\]
