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 defined by:

\[ f(x) = \int \frac{5x^4 + 4x^5}{(x^5 + x + 1)^2} \, dx \]

with the condition that \( f(0) = 0 \), and we want to find \( f(1) \).

### Step 1: Simplifying the Integral

We start by simplifying the integrand:

\[ f(x) = \int \frac{5x^4 + 4x^5}{(x^5 + x + 1)^2} \, dx \]

We can factor out \( x^4 \) from the numerator:

\[ f(x) = \int \frac{x^4(5 + 4x)}{(x^5 + x + 1)^2} \, dx \]

### Step 2: Substitution

Next, we will use substitution to simplify the integral. Let:

\[ t = x^5 + x + 1 \]

Then, we differentiate \( t \) with respect to \( x \):

\[ dt = (5x^4 + 1) \, dx \Rightarrow dx = \frac{dt}{5x^4 + 1} \]

### Step 3: Rewriting the Integral

Now we need to express \( 5x^4 + 4x^5 \) in terms of \( t \):

\[ 5x^4 + 4x^5 = 5x^4 + 4(x^5) = 5x^4 + 4(t - x - 1) \]

Substituting this into the integral gives:

\[ f(x) = \int \frac{5x^4 + 4(t - x - 1)}{t^2} \cdot \frac{dt}{5x^4 + 1} \]

### Step 4: Evaluating the Integral

This integral can be simplified further, but we can also evaluate it directly for specific values of \( x \).

### Step 5: Finding \( f(0) \)

Given that \( f(0) = 0 \), we can find the constant of integration \( C \) after evaluating the integral.

### Step 6: Finding \( f(1) \)

Now we calculate \( f(1) \):

\[ f(1) = \int \frac{5(1)^4 + 4(1)^5}{(1^5 + 1 + 1)^2} \, dx = \int \frac{5 + 4}{(1 + 1 + 1)^2} \, dx = \int \frac{9}{9} \, dx = \int 1 \, dx \]

Evaluating this integral from 0 to 1 gives:

\[ f(1) = 1 - 0 = 1 \]

### Conclusion

Thus, the value of \( f(1) \) is:

\[ \boxed{1} \]