If `overset(b)underset(a)int (x^(n))/(x^(4)+(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 given integral equation

\[ \int_a^b \frac{x^n}{x^4 + (16 - x)^n} \, dx = 6, \]

we will use the property of definite integrals.

### Step 1: Use the property of definite integrals

We know that

\[ \int_a^b f(x) \, dx = \int_a^b f(a + b - x) \, dx. \]

Let \( a + b = 16 \). Then, we can rewrite the integral as:

\[ I = \int_a^b \frac{x^n}{x^4 + (16 - x)^n} \, dx. \]

Using the property, we have:

\[ I = \int_a^b \frac{(16 - x)^n}{(16 - x)^4 + x^n} \, dx. \]

### Step 2: Add the two integrals

Now, we can add the two expressions for \( I \):

\[ 2I = \int_a^b \left( \frac{x^n}{x^4 + (16 - x)^n} + \frac{(16 - x)^n}{(16 - x)^4 + x^n} \right) \, dx. \]

### Step 3: Simplify the integrand

The integrand simplifies to:

\[ \frac{x^n + (16 - x)^n}{x^4 + (16 - x)^n}. \]

Thus, we have:

\[ 2I = \int_a^b \frac{x^n + (16 - x)^n}{x^4 + (16 - x)^n} \, dx. \]

### Step 4: Evaluate the integral

Now, we can evaluate the integral:

\[ 2I = \int_a^b dx = b - a. \]

### Step 5: Relate \( b - a \) to \( I \)

Since we know that \( I = 6 \), we have:

\[ 2I = 12. \]

Thus, we can write:

\[ b - a = 12. \]

### Step 6: Solve the system of equations

Now we have two equations:

1. \( a + b = 16 \)
2. \( b - a = 12 \)

We can solve these equations simultaneously.

Adding the two equations:

\[ (a + b) + (b - a) = 16 + 12 \implies 2b = 28 \implies b = 14. \]

Substituting \( b = 14 \) into the first equation:

\[ a + 14 = 16 \implies a = 2. \]

### Conclusion

Thus, we find that:

\[ a = 2, \quad b = 14. \]

The final answer is:

**Option B: \( a = 2, b = 14, n \in \mathbb{R} \)**.

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