Let `int(x^(3)+x^(2)+x)/(sqrt(12x^(3)+15x^(2)+20x))dx=f(x)` where `f(1)=(sqrt(47))/(30).` If `(f(2))^(2)` is equal to `(p)/(255)`, then the value of p is equal to

Answer

To solve the problem, we need to evaluate the integral given and find the value of \( p \) such that \( (f(2))^2 = \frac{p}{255} \).

### Step-by-Step Solution:

1. **Rewrite the Integral**:
\[ f(x) = \int \frac{x^3 + x^2 + x}{\sqrt{12x^3 + 15x^2 + 20x}} \, dx \]
We can multiply the numerator and denominator by \( x \):
\[ f(x) = \int \frac{x^4 + x^3 + x^2}{\sqrt{12x^5 + 15x^4 + 20x^2}} \, dx \]

2. **Substitution**:
Let:
\[ t = 12x^5 + 15x^4 + 20x^2 \]
Then, differentiate \( t \):
\[ dt = (60x^4 + 60x^3 + 40x) \, dx \]
Rearranging gives:
\[ dx = \frac{dt}{60(x^4 + x^3 + \frac{2}{3}x)} \]

3. **Substituting in the Integral**:
Substitute \( t \) into the integral:
\[ f(x) = \int \frac{(x^4 + x^3 + x^2)}{\sqrt{t}} \cdot \frac{dt}{60(x^4 + x^3 + \frac{2}{3}x)} \]
Simplifying gives:
\[ f(x) = \frac{1}{60} \int \frac{dt}{\sqrt{t}} \]

4. **Integrate**:
The integral of \( \frac{1}{\sqrt{t}} \) is:
\[ \int t^{-1/2} dt = 2\sqrt{t} + C \]
Therefore:
\[ f(x) = \frac{1}{60} \cdot 2\sqrt{t} + C = \frac{1}{30} \sqrt{12x^5 + 15x^4 + 20x^2} + C \]

5. **Finding the Constant \( C \)**:
We know \( f(1) = \frac{\sqrt{47}}{30} \):
\[ f(1) = \frac{1}{30} \sqrt{12(1)^5 + 15(1)^4 + 20(1)^2} + C \]
Simplifying gives:
\[ f(1) = \frac{1}{30} \sqrt{12 + 15 + 20} + C = \frac{1}{30} \sqrt{47} + C \]
Setting this equal to \( \frac{\sqrt{47}}{30} \):
\[ C = 0 \]
Thus:
\[ f(x) = \frac{1}{30} \sqrt{12x^5 + 15x^4 + 20x^2} \]

6. **Finding \( f(2) \)**:
Now we compute \( f(2) \):
\[ f(2) = \frac{1}{30} \sqrt{12(2^5) + 15(2^4) + 20(2^2)} \]
Calculating inside the square root:
\[ = \frac{1}{30} \sqrt{12 \cdot 32 + 15 \cdot 16 + 20 \cdot 4} = \frac{1}{30} \sqrt{384 + 240 + 80} = \frac{1}{30} \sqrt{704} \]

7. **Calculating \( (f(2))^2 \)**:
\[ (f(2))^2 = \left(\frac{1}{30} \sqrt{704}\right)^2 = \frac{704}{900} \]

8. **Setting up the equation**:
We know that:
\[ (f(2))^2 = \frac{p}{255} \]
Thus:
\[ \frac{704}{900} = \frac{p}{255} \]

9. **Cross-multiplying**:
\[ p = \frac{704 \cdot 255}{900} \]

10. **Calculating \( p \)**:
Simplifying:
\[ p = \frac{704 \cdot 255}{900} = \frac{704 \cdot 255}{900} = 196 \]

### Final Answer:
The value of \( p \) is \( 196 \).