Consider `int(3x^(4)+2x^(2)+1)/(sqrt(x^(4)+x^(2)+1))dx=f(x)`. If `f(1)=sqrt3`, then `(f(2))^(2)` is equal to

Answer

To solve the given problem, we need to evaluate the integral and find the function \( f(x) \) based on the provided information. Let's go through the steps one by one.

### Step 1: Set up the integral
We start with the integral:
\[ f(x) = \int \frac{3x^4 + 2x^2 + 1}{\sqrt{x^4 + x^2 + 1}} \, dx \]

### Step 2: Simplify the integrand
To simplify the integrand, we can multiply and divide by \( x \):
\[ f(x) = \int \frac{(3x^4 + 2x^2 + 1)x}{x \sqrt{x^4 + x^2 + 1}} \, dx = \int \frac{3x^5 + 2x^3 + x}{\sqrt{x^6 + x^4 + x^2}} \, dx \]

### Step 3: Substitute for easier integration
Let:
\[ T = x^6 + x^4 + x^2 \]
Then, differentiate \( T \):
\[ \frac{dT}{dx} = 6x^5 + 4x^3 + 2x \]
Thus, we can express \( dx \) in terms of \( dT \):
\[ dx = \frac{dT}{6x^5 + 4x^3 + 2x} \]

### Step 4: Rewrite the integral in terms of \( T \)
Now, we can rewrite the integral:
\[ f(x) = \int \frac{1}{\sqrt{T}} \cdot \frac{dT}{6x^5 + 4x^3 + 2x} \]

### Step 5: Integrate
The integral of \( \frac{1}{\sqrt{T}} \) is:
\[ \int \frac{1}{\sqrt{T}} \, dT = 2\sqrt{T} + C \]
Thus,
\[ f(x) = 2\sqrt{x^6 + x^4 + x^2} + C \]

### Step 6: Use the given condition to find \( C \)
We know that \( f(1) = \sqrt{3} \):
\[ f(1) = 2\sqrt{1^6 + 1^4 + 1^2} + C = 2\sqrt{3} + C \]
Setting this equal to \( \sqrt{3} \):
\[ 2\sqrt{3} + C = \sqrt{3} \]
Thus, solving for \( C \):
\[ C = \sqrt{3} - 2\sqrt{3} = -\sqrt{3} \]

### Step 7: Write the final expression for \( f(x) \)
Now we have:
\[ f(x) = 2\sqrt{x^6 + x^4 + x^2} - \sqrt{3} \]

### Step 8: Calculate \( f(2) \)
Now we need to find \( f(2) \):
\[ f(2) = 2\sqrt{2^6 + 2^4 + 2^2} - \sqrt{3} \]
Calculating the terms:
\[ 2^6 = 64, \quad 2^4 = 16, \quad 2^2 = 4 \]
Thus,
\[ f(2) = 2\sqrt{64 + 16 + 4} - \sqrt{3} = 2\sqrt{84} - \sqrt{3} \]

### Step 9: Find \( (f(2))^2 \)
Now we compute \( (f(2))^2 \):
\[ (f(2))^2 = (2\sqrt{84} - \sqrt{3})^2 = 4 \cdot 84 + 3 - 8\sqrt{84}\sqrt{3} \]
Calculating:
\[ = 336 + 3 - 8\sqrt{252} = 339 - 8\sqrt{252} \]

### Final Answer
Thus, the value of \( (f(2))^2 \) is:
\[ \boxed{336} \]