If `int_(a)^(b) (x^(n))/(x^(n)+(16-x)^(n))dx=6`, then

Options

A. `a=4,b=12,n in R`
B. `a=2,b=14,n in R`
C. `a=-4,b=20,n in R`
D. `a=2,b=8, n in R`

Answer

To solve the integral equation given in the problem, we will follow a systematic approach using properties of definite integrals.

### Step-by-step Solution:

1. **Given Integral**: We start with the integral equation:
\[ I = \int_{a}^{b} \frac{x^n}{x^n + (16 - x)^n} \, dx = 6 \]

2. **Use the Property of Integrals**: We can use the property of integrals that states:
\[ \int_{a}^{b} f(x) \, dx = \int_{a}^{b} f(a + b - x) \, dx \]
Here, let \( a + b = 16 \). Therefore, we can rewrite the integral as:
\[ I = \int_{a}^{b} \frac{(16 - x)^n}{(16 - x)^n + x^n} \, dx \]

3. **Combine the Two Integrals**: Now we have two expressions for \( I \):
\[ I = \int_{a}^{b} \frac{x^n}{x^n + (16 - x)^n} \, dx \]
and
\[ I = \int_{a}^{b} \frac{(16 - x)^n}{(16 - x)^n + x^n} \, dx \]
Adding these two equations gives:
\[ 2I = \int_{a}^{b} \left( \frac{x^n}{x^n + (16 - x)^n} + \frac{(16 - x)^n}{(16 - x)^n + x^n} \right) \, dx \]

4. **Simplifying the Expression**: The sum inside the integral simplifies to:
\[ 2I = \int_{a}^{b} 1 \, dx \]
This is because the two fractions add up to 1.

5. **Evaluate the Integral**: Now, we can evaluate the integral:
\[ 2I = \int_{a}^{b} 1 \, dx = b - a \]

6. **Substituting the Value of I**: We know from the problem statement that \( I = 6 \):
\[ 2 \cdot 6 = b - a \implies 12 = b - a \]

7. **Using the Sum of Limits**: We also have the equation from our assumption:
\[ a + b = 16 \]

8. **Solving the System of Equations**: Now we have a system of equations:
\[ b - a = 12 \quad \text{(1)} \]
\[ a + b = 16 \quad \text{(2)} \]
From equation (1), we can express \( b \) in terms of \( a \):
\[ b = a + 12 \]
Substituting this into equation (2):
\[ a + (a + 12) = 16 \implies 2a + 12 = 16 \implies 2a = 4 \implies a = 2 \]
Now substituting \( a = 2 \) back into \( b = a + 12 \):
\[ b = 2 + 12 = 14 \]

9. **Final Values**: Thus, the values of \( a \) and \( b \) are:
\[ a = 2, \quad b = 14 \]

### Conclusion:
The values of \( a \) and \( b \) that satisfy the integral equation are \( a = 2 \) and \( b = 14 \).