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 solve the problem of finding 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 area between the two curves, we first need to determine where they intersect. We set the equations equal to each other:

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

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

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

This simplifies to:

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

Let \( k = x^2 \). Then we have:

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

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

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

This gives us:

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

Since \( k = x^2 \) must be non-negative, we take \( k = 1 \). Thus, \( x^2 = 1 \) 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:

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

The first integral is:

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

Evaluating from \( -1 \) to \( 1 \):

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

The second integral is:

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

Evaluating from \( -1 \) to \( 1 \):

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

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

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

### Step 6: Identify \( K_1 \) and \( K_2 \)
We have the area in the form \( K_1 \pi - \frac{K_2}{3} \). Here, \( K_1 = 1 \) and \( K_2 = 2 \).

### Step 7: Calculate \( K_1 + K_2 \)
Thus, we find:

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

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