If the value of integral
`int(x+sqrt(x^2 -1))^(2)dx = ax^3 - x + b(x^2 - 1)^(1/b), +C`
(where, C is the constant of integration), then `a xx b` is equal to

Options

A. 1
B. `4/9`
C. `2`
D. `9/4`

Answer

To solve the integral \( \int (x + \sqrt{x^2 - 1})^2 \, dx \) and express it in the form \( ax^3 - x + b(x^2 - 1)^{1/b} + C \), we will follow these steps:

### Step 1: Expand the integrand
First, we need to expand the expression \( (x + \sqrt{x^2 - 1})^2 \):
\[ (x + \sqrt{x^2 - 1})^2 = x^2 + 2x\sqrt{x^2 - 1} + (x^2 - 1) = 2x^2 - 1 + 2x\sqrt{x^2 - 1} \]

### Step 2: Set up the integral
Now, we can set up the integral:
\[ \int (2x^2 - 1 + 2x\sqrt{x^2 - 1}) \, dx \]
This can be separated into three integrals:
\[ \int (2x^2) \, dx - \int 1 \, dx + \int (2x\sqrt{x^2 - 1}) \, dx \]

### Step 3: Compute the first two integrals
The first integral:
\[ \int 2x^2 \, dx = \frac{2}{3}x^3 \]
The second integral:
\[ \int 1 \, dx = x \]

### Step 4: Compute the third integral
For the third integral \( \int 2x\sqrt{x^2 - 1} \, dx \), we can use substitution. Let \( u = x^2 - 1 \), then \( du = 2x \, dx \). Thus, we have:
\[ \int 2x\sqrt{x^2 - 1} \, dx = \int \sqrt{u} \, du = \frac{2}{3}u^{3/2} + C = \frac{2}{3}(x^2 - 1)^{3/2} \]

### Step 5: Combine the results
Now, combining all the parts, we get:
\[ \int (x + \sqrt{x^2 - 1})^2 \, dx = \frac{2}{3}x^3 - x + \frac{2}{3}(x^2 - 1)^{3/2} + C \]

### Step 6: Identify coefficients
From the expression, we can identify:
- \( a = \frac{2}{3} \)
- \( b = \frac{2}{3} \)

### Step 7: Calculate \( a \times b \)
Now, we calculate \( a \times b \):
\[ a \times b = \left(\frac{2}{3}\right) \times \left(\frac{2}{3}\right) = \frac{4}{9} \]

Thus, the final answer is:
\[ \boxed{\frac{4}{9}} \]