If `f(x)=int((5x^4+4x^5))/((x^5+x+1)^(2))dx` and f(0)=0, then the value of f(1) is :

Options

A. 1
B. `(1)/(2)`
C. `(1)/(3)`
D. `(1)/(5)`

Answer

To solve the problem, we need to evaluate the integral function \( f(x) = \int \frac{5x^4 + 4x^5}{(x^5 + x + 1)^2} \, dx \) and find \( f(1) \) given that \( f(0) = 0 \).

### Step-by-Step Solution

1. **Rewrite the Integral**:
\[ f(x) = \int \frac{5x^4 + 4x^5}{(x^5 + x + 1)^2} \, dx \]

2. **Factor Out Common Terms**:
In the numerator, we can factor out \( x^4 \):
\[ f(x) = \int \frac{x^4(5 + 4x)}{(x^5 + x + 1)^2} \, dx \]

3. **Substitution**:
Let \( t = x^5 + x + 1 \). Then, we differentiate \( t \):
\[ dt = (5x^4 + 1) \, dx \quad \Rightarrow \quad dx = \frac{dt}{5x^4 + 1} \]

4. **Rearranging the Integral**:
Substitute \( t \) into the integral:
\[ f(x) = \int \frac{x^4(5 + 4x)}{t^2} \cdot \frac{dt}{5x^4 + 1} \]

5. **Simplifying the Integral**:
The integral simplifies to:
\[ f(x) = \int \frac{(5 + 4x)}{t^2} \, dt \]

6. **Integrate**:
The integral of \( t^{-2} \) is:
\[ \int t^{-2} \, dt = -\frac{1}{t} + C \]

7. **Substituting Back**:
Substitute back \( t = x^5 + x + 1 \):
\[ f(x) = -\frac{1}{x^5 + x + 1} + C \]

8. **Using the Condition \( f(0) = 0 \)**:
We know \( f(0) = 0 \):
\[ f(0) = -\frac{1}{0^5 + 0 + 1} + C = -1 + C = 0 \quad \Rightarrow \quad C = 1 \]

9. **Final Expression for \( f(x) \)**:
Thus, we have:
\[ f(x) = -\frac{1}{x^5 + x + 1} + 1 \]

10. **Evaluate \( f(1) \)**:
Now, we find \( f(1) \):
\[ f(1) = -\frac{1}{1^5 + 1 + 1} + 1 = -\frac{1}{3} + 1 = 1 - \frac{1}{3} = \frac{2}{3} \]

### Conclusion
The value of \( f(1) \) is \( \frac{2}{3} \).