`int (x^(2))/((1-x^(6)))dx=?`

Options

A. `(1)/(6)log |(1+x^(3))/(1-x^(3))|+C`
B. `(1)/(6)log |(1-x^(3))/(1+x^(3))|+C`
C. `(1)/(3)log |(1-x^(3))/(1+x^(3))|+C`
D. None of these

Explanation

To solve the integral \( \int \frac{x^2}{1 - x^6} \, dx \), we can follow these steps:

### Step 1: Rewrite the Integral
We start with the integral:
\[ I = \int \frac{x^2}{1 - x^6} \, dx \]

### Step 2: Factor the Denominator
Notice that \( 1 - x^6 \) can be factored as:
\[ 1 - x^6 = (1 - x^3)(1 + x^3) \]
However, for our purpose, we will use the identity directly as it is.

### Step 3: Use Substitution
Let \( t = x^3 \). Then, the differential \( dt \) is given by:
\[ dt = 3x^2 \, dx \quad \Rightarrow \quad dx = \frac{dt}{3x^2} \]
Substituting \( x^2 \, dx \) gives us:
\[ x^2 \, dx = \frac{1}{3} dt \]

### Step 4: Substitute into the Integral
Now substitute \( t \) into the integral:
\[ I = \int \frac{x^2}{1 - x^6} \, dx = \int \frac{\frac{1}{3} dt}{1 - t^2} \]
This simplifies to:
\[ I = \frac{1}{3} \int \frac{dt}{1 - t^2} \]

### Step 5: Recognize the Integral Form
The integral \( \int \frac{dt}{1 - t^2} \) can be solved using the formula:
\[ \int \frac{dt}{1 - t^2} = \frac{1}{2} \log \left| \frac{1 + t}{1 - t} \right| + C \]

### Step 6: Substitute Back
Now substitute back:
\[ I = \frac{1}{3} \cdot \frac{1}{2} \log \left| \frac{1 + t}{1 - t} \right| + C = \frac{1}{6} \log \left| \frac{1 + x^3}{1 - x^3} \right| + C \]

### Final Answer
Thus, the final result of the integral is:
\[ \int \frac{x^2}{1 - x^6} \, dx = \frac{1}{6} \log \left| \frac{1 + x^3}{1 - x^3} \right| + C \]