If the area bounded by `y=x^(2) and y=(2)/(1+x^(2))` is `(K_(1)pi-(K_(2))/(3))` sq. units (where `K_(1), K_(2) in Z`), then the value of `(K_(1)+K_(2))` is equal to

Options

A. 3
B. 1
C. `-1`
D. `-2`

Answer

To find the area bounded by the curves \( y = x^2 \) and \( y = \frac{2}{1 + x^2} \), we will follow these steps:

### Step 1: Find the points of intersection
To find the points of intersection, we set the two equations equal to each other:

\[ x^2 = \frac{2}{1 + x^2} \]

Multiplying both sides by \( 1 + x^2 \) to eliminate the fraction, we get:

\[ x^2(1 + x^2) = 2 \]

This simplifies to:

\[ x^4 + x^2 - 2 = 0 \]

Let \( k = x^2 \). Then we can rewrite the equation as:

\[ k^2 + k - 2 = 0 \]

### Step 2: Solve the quadratic equation
Using the quadratic formula \( k = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \):

\[ k = \frac{-1 \pm \sqrt{1^2 - 4 \cdot 1 \cdot (-2)}}{2 \cdot 1} = \frac{-1 \pm \sqrt{1 + 8}}{2} = \frac{-1 \pm 3}{2} \]

This gives us:

\[ k = 1 \quad \text{or} \quad k = -2 \]

Since \( k = x^2 \), we discard \( k = -2 \) and take \( k = 1 \). Therefore, \( x^2 = 1 \) implies \( x = 1 \) and \( x = -1 \).

### Step 3: Set up the integral for the area
The area \( A \) between the curves from \( x = -1 \) to \( x = 1 \) is given by:

\[ A = \int_{-1}^{1} \left( \frac{2}{1 + x^2} - x^2 \right) dx \]

### Step 4: Evaluate the integral
We can split the integral:

\[ A = \int_{-1}^{1} \frac{2}{1 + x^2} \, dx - \int_{-1}^{1} x^2 \, dx \]

1. **First integral**:

\[ \int \frac{2}{1 + x^2} \, dx = 2 \tan^{-1}(x) \]

Evaluating from -1 to 1:

\[ \left[ 2 \tan^{-1}(x) \right]_{-1}^{1} = 2 \left( \tan^{-1}(1) - \tan^{-1}(-1) \right) = 2 \left( \frac{\pi}{4} - \left(-\frac{\pi}{4}\right) \right) = 2 \cdot \frac{\pi}{2} = \pi \]

2. **Second integral**:

\[ \int x^2 \, dx = \frac{x^3}{3} \]

Evaluating from -1 to 1:

\[ \left[ \frac{x^3}{3} \right]_{-1}^{1} = \frac{1^3}{3} - \frac{(-1)^3}{3} = \frac{1}{3} - \left(-\frac{1}{3}\right) = \frac{2}{3} \]

### Step 5: Combine the results
Now, substituting back into the area formula:

\[ A = \pi - \frac{2}{3} \]

### Step 6: Express in the required form
The area is given in the form \( K_1 \pi - \frac{K_2}{3} \). Here, we have:

\[ K_1 = 1 \quad \text{and} \quad K_2 = 2 \]

### Final Step: Calculate \( K_1 + K_2 \)
Thus,

\[ K_1 + K_2 = 1 + 2 = 3 \]

### Answer
The value of \( K_1 + K_2 \) is \( \boxed{3} \).

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