Let `f(x)=(ax^2+bx+c)/(x^2+1)` such that y=-2 is an asymptote of the curve `y=f(x)`. The curve `y=f(x)` is symmetric about Y-axis and its maximum values is 4. Let `h(x)=f(x)-g(x)`,where `f(x)=sin^4 pi x` and `g(x)=log_(e)x`. Let `x_(0),x_(1),x_(2)...x_(n+1)` be the roots of `f(x)=g(x)` in increasing order
Then, the absolute area enclosed by `y=f(x)` and `y=g(x)` is given by

Options

A. `sum_(r=0)^n int_(x_r)^(x_(r+1))(-1)^rcdoth(x)dx`
B. `sum_(r=0)^n int_(x_1)^(x_(r+1))(-1)^(r+1)cdot h(x)dx`
C. `2sum_(r=0)^n int_(x_r)^(x_(r__r+1))(-1)^rcdoth(x)dx`
D. `1/2 cdot sum_(r=0)^n int_(x_1)^(x_(r+1))(-1)^(r+1)cdot h(x)dx`

Answer

To solve the given problem, we will follow these steps:

### Step 1: Understanding the Function and Asymptote
Given the function:
\[ f(x) = \frac{ax^2 + bx + c}{x^2 + 1} \]
We know that \( y = -2 \) is an asymptote. This means that as \( x \) approaches infinity, \( f(x) \) approaches -2. Therefore, we can set up the equation:
\[ \lim_{x \to \infty} f(x) = -2 \]
This implies:
\[ \lim_{x \to \infty} \frac{ax^2 + bx + c}{x^2 + 1} = \frac{a}{1} = -2 \]
Thus, we find:
\[ a = -2 \]

### Step 2: Symmetry and Maximum Value
The function is symmetric about the Y-axis, which means \( b = 0 \) (the odd function term must vanish). Therefore, we have:
\[ f(x) = \frac{-2x^2 + c}{x^2 + 1} \]

Next, we need to find the maximum value of \( f(x) \). The maximum value is given as 4. To find the maximum, we can set the derivative \( f'(x) \) to zero and solve for critical points.

### Step 3: Finding Maximum Value
To find the maximum value, we first differentiate \( f(x) \):
Using the quotient rule:
\[ f'(x) = \frac{(x^2 + 1)(-4x) - (-2x^2 + c)(2x)}{(x^2 + 1)^2} \]
Setting \( f'(x) = 0 \) leads us to find the critical points.

After finding the critical points, we substitute back into \( f(x) \) to find the maximum value. Setting this equal to 4 gives us:
\[ \frac{-2x^2 + c}{x^2 + 1} = 4 \]
Solving this equation will give us the value of \( c \).

### Step 4: Setting Up the Functions \( h(x) \)
Now we define:
\[ h(x) = f(x) - g(x) \]
Where:
\[ g(x) = \sin^4(\pi x) - \log_e(x) \]

### Step 5: Finding Roots
We need to find the roots of the equation \( f(x) = g(x) \). Let \( x_0, x_1, x_2, \ldots, x_{n+1} \) be the roots in increasing order.

### Step 6: Area Calculation
The area enclosed by the curves \( y = f(x) \) and \( y = g(x) \) can be calculated using the definite integrals between the roots:
\[ \text{Area} = \int_{x_0}^{x_1} |f(x) - g(x)| \, dx + \int_{x_1}^{x_2} |f(x) - g(x)| \, dx + \ldots + \int_{x_n}^{x_{n+1}} |f(x) - g(x)| \, dx \]

### Step 7: Final Expression
The total area can be expressed as:
\[ \text{Area} = \sum_{r=0}^{n} \int_{x_r}^{x_{r+1}} |f(x) - g(x)| \, dx \]