If `int_((-1)/(sqrt3))^(1//sqrt3)(x^(4))/(1-x^(4))cos^(-1)((2x)/(1+x^(2)))dx=k`, then `int_(0)^(1//sqrt3)(x^(4))/(1-x^(4))dx`
the value of k is equal to

Options

A. `pi`
B. `2pi`
C. `-pi`
D. `3pi`

Answer

To solve the given integral problem, we will follow a structured approach step by step.

### Step 1: Define the Integral
We start with the integral:
\[ I = \int_{-\frac{1}{\sqrt{3}}}^{\frac{1}{\sqrt{3}}} \frac{x^4}{1 - x^4} \cos^{-1}\left(\frac{2x}{1 + x^2}\right) dx \]

### Step 2: Use the Identity for Cosine Inverse
We can use the identity:
\[ \cos^{-1}(x) = \frac{\pi}{2} - \sin^{-1}(x) \]
Thus, we rewrite the integral:
\[ I = \int_{-\frac{1}{\sqrt{3}}}^{\frac{1}{\sqrt{3}}} \frac{x^4}{1 - x^4} \left(\frac{\pi}{2} - \sin^{-1}\left(\frac{2x}{1 + x^2}\right)\right) dx \]

### Step 3: Split the Integral
Now, we can split the integral into two parts:
\[ I = \int_{-\frac{1}{\sqrt{3}}}^{\frac{1}{\sqrt{3}}} \frac{x^4}{1 - x^4} \cdot \frac{\pi}{2} \, dx - \int_{-\frac{1}{\sqrt{3}}}^{\frac{1}{\sqrt{3}}} \frac{x^4}{1 - x^4} \sin^{-1}\left(\frac{2x}{1 + x^2}\right) dx \]

### Step 4: Analyze the Functions
Next, we analyze the symmetry of the integrands:
- The function \( \frac{x^4}{1 - x^4} \) is even because replacing \( x \) with \( -x \) does not change the function.
- The function \( \sin^{-1}\left(\frac{2x}{1 + x^2}\right) \) is odd because replacing \( x \) with \( -x \) gives \( -\sin^{-1}\left(\frac{2x}{1 + x^2}\right) \).

### Step 5: Apply the Properties of Integrals
Using the properties of integrals:
- The integral of an odd function over a symmetric interval around zero is zero.
Thus, the second integral becomes:
\[ \int_{-\frac{1}{\sqrt{3}}}^{\frac{1}{\sqrt{3}}} \frac{x^4}{1 - x^4} \sin^{-1}\left(\frac{2x}{1 + x^2}\right) dx = 0 \]

### Step 6: Simplify the Integral
Now, we are left with:
\[ I = \frac{\pi}{2} \int_{-\frac{1}{\sqrt{3}}}^{\frac{1}{\sqrt{3}}} \frac{x^4}{1 - x^4} \, dx \]
Using the property of even functions, we can double the integral from \( 0 \) to \( \frac{1}{\sqrt{3}} \):
\[ I = \pi \int_{0}^{\frac{1}{\sqrt{3}}} \frac{x^4}{1 - x^4} \, dx \]

### Step 7: Relate to k
From the problem statement, we know that:
\[ I = k \]
Thus, we have:
\[ k = \pi \int_{0}^{\frac{1}{\sqrt{3}}} \frac{x^4}{1 - x^4} \, dx \]

### Conclusion
The value of \( k \) is:
\[ k = \pi \]