The coefficient of `x^(6)` in the expansion of
`log{(1+x)^(1+x)(1-x)^(1-x)`}` is

Options

A. `(1)/(15)`
B. `(1)/(30)`
C. `(1)/(10)`
D. `(1)/(45)`

Answer

To find the coefficient of \( x^6 \) in the expansion of \( \log{(1+x)^{1+x}(1-x)^{1-x}} \), we can follow these steps:

### Step 1: Rewrite the Expression
We start with the expression:
\[ \log{(1+x)^{1+x}(1-x)^{1-x}} \]
Using the logarithmic property \( \log{(a \cdot b)} = \log{a} + \log{b} \), we can rewrite this as:
\[ \log{(1+x)^{1+x}} + \log{(1-x)^{1-x}} \]

### Step 2: Apply Logarithmic Properties
Using the property \( \log{(a^b)} = b \log{a} \), we can further simplify:
\[ (1+x) \log{(1+x)} + (1-x) \log{(1-x)} \]

### Step 3: Expand the Logarithms
Now, we need the series expansions for \( \log{(1+x)} \) and \( \log{(1-x)} \):
\[ \log{(1+x)} = x - \frac{x^2}{2} + \frac{x^3}{3} - \frac{x^4}{4} + \frac{x^5}{5} - \frac{x^6}{6} + \ldots \]
\[ \log{(1-x)} = -\left(x + \frac{x^2}{2} + \frac{x^3}{3} + \frac{x^4}{4} + \frac{x^5}{5} + \frac{x^6}{6} + \ldots\right) \]

### Step 4: Substitute the Expansions
Substituting these expansions back into our expression gives:
\[ (1+x) \left(x - \frac{x^2}{2} + \frac{x^3}{3} - \frac{x^4}{4} + \frac{x^5}{5} - \frac{x^6}{6} + \ldots\right) + (1-x) \left(-x - \frac{x^2}{2} - \frac{x^3}{3} - \frac{x^4}{4} - \frac{x^5}{5} - \frac{x^6}{6} - \ldots\right) \]

### Step 5: Combine the Terms
Now, we need to combine the terms to find the coefficient of \( x^6 \):
1. From \( (1+x) \):
- \( 1 \cdot \left(-\frac{x^6}{6}\right) = -\frac{1}{6} \)
- \( x \cdot \left(\frac{x^5}{5}\right) = \frac{1}{5} \)

2. From \( (1-x) \):
- \( 1 \cdot (-x) \cdot \left(-\frac{x^5}{5}\right) = -\frac{1}{6} \)
- \( -x \cdot \left(-\frac{x^5}{5}\right) = \frac{1}{5} \)

### Step 6: Calculate the Coefficient
Combining all these contributions:
\[ -\frac{1}{6} + \frac{1}{5} - \frac{1}{6} + \frac{1}{5} \]
Combining like terms:
\[ -\frac{2}{6} + \frac{2}{5} = -\frac{1}{3} + \frac{2}{5} \]
Finding a common denominator (15):
\[ -\frac{5}{15} + \frac{6}{15} = \frac{1}{15} \]

### Conclusion
The coefficient of \( x^6 \) in the expansion is:
\[ \frac{1}{15} \]