If `f(x) = int(5x^(8)+7x^(6))/((x^(2)+1+2x^(7))^(2))dx, (x ge 0)`, and f(0) = 0, then the value of f(1) is

Options

A. `-(1)/(2)`
B. `-(1)/(4)`
C. `(1)/(4)`
D. `(1)/(2)`

Explanation

To solve the problem, we need to evaluate the integral

\[ f(x) = \int \frac{5x^8 + 7x^6}{(x^2 + 1 + 2x^7)^2} \, dx \]

with the condition that \( f(0) = 0 \) and find \( f(1) \).

### Step 1: Simplifying the Integral

We start by simplifying the integrand. Notice that we can factor out \( x^7 \) from the denominator:

\[ f(x) = \int \frac{5x^8 + 7x^6}{(x^2 + 1 + 2x^7)^2} \, dx \]

Taking \( x^7 \) out of the denominator gives us:

\[ = \int \frac{5x^8 + 7x^6}{x^{14} \left(\frac{x^2}{x^7} + \frac{1}{x^7} + 2\right)^2} \, dx \]

This simplifies to:

\[ = \int \frac{5x^8 + 7x^6}{x^{14} \left(x^{-5} + x^{-7} + 2\right)^2} \, dx \]

### Step 2: Changing Variables

Let \( t = x^{-5} + x^{-7} + 2 \). We need to find \( dt \):

\[ dt = \left(-5x^{-6} - 7x^{-8}\right) dx \]

Thus,

\[ dx = \frac{dt}{-5x^{-6} - 7x^{-8}} \]

### Step 3: Substituting in the Integral

Substituting \( t \) into the integral, we have:

\[ f(t) = \int \frac{-dt}{t^2} \]

### Step 4: Integrating

Now we can integrate:

\[ f(t) = \int -t^{-2} \, dt = \frac{1}{t} + C \]

### Step 5: Back Substituting

Now we substitute back for \( t \):

\[ f(x) = \frac{1}{x^{-5} + x^{-7} + 2} + C \]

### Step 6: Applying the Condition \( f(0) = 0 \)

To find \( C \), we use the condition \( f(0) = 0 \):

\[ f(0) = \frac{1}{0 + 0 + 2} + C = 0 \implies C = -\frac{1}{2} \]

Thus,

\[ f(x) = \frac{1}{x^{-5} + x^{-7} + 2} - \frac{1}{2} \]

### Step 7: Evaluating \( f(1) \)

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

\[ f(1) = \frac{1}{1^{-5} + 1^{-7} + 2} - \frac{1}{2} \]

Calculating the terms:

\[ = \frac{1}{1 + 1 + 2} - \frac{1}{2} = \frac{1}{4} - \frac{1}{2} = \frac{1}{4} - \frac{2}{4} = -\frac{1}{4} \]

### Final Result

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

\[ f(1) = \frac{1}{4} \]