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. `1/2`
C. `-1/4`
D. `1/4`

Answer

To solve the problem step by step, we need to evaluate the integral given in the function \( f(x) \) and find \( f(1) \).

### Step 1: Write down the integral
The function is defined as:
\[ f(x) = \int \frac{5x^8 + 7x^6}{(x^2 + 1 + 2x^7)^2} \, dx \]

### Step 2: Simplify the integrand
Notice that we can factor out \( x^7 \) from the denominator:
\[ x^2 + 1 + 2x^7 = x^7 \left( \frac{x^2}{x^7} + \frac{1}{x^7} + 2 \right) = x^7 \left( \frac{1}{x^5} + \frac{1}{x^7} + 2 \right) \]
Thus, the integrand becomes:
\[ \frac{5x^8 + 7x^6}{(x^2 + 1 + 2x^7)^2} = \frac{5x^8 + 7x^6}{x^{14} \left( \frac{1}{x^5} + \frac{1}{x^7} + 2 \right)^2} \]

### Step 3: Change of variables
Let \( t = x^{-5} + x^{-7} + 2 \). Then we differentiate:
\[ \frac{dt}{dx} = -5x^{-6} - 7x^{-8} \]
Thus,
\[ dx = -\frac{dt}{5x^{-6} + 7x^{-8}} \]

### Step 4: Substitute in the integral
Now we can substitute \( t \) into the integral:
\[ f(x) = \int -\frac{1}{t^2} \, dt \]

### Step 5: Evaluate the integral
The integral of \( -\frac{1}{t^2} \) is:
\[ \int -\frac{1}{t^2} \, dt = \frac{1}{t} + C \]

### Step 6: Substitute back for \( t \)
Substituting back for \( t \):
\[ f(x) = \frac{1}{x^{-5} + x^{-7} + 2} + C \]

### Step 7: Use the condition \( f(0) = 0 \)
We know \( f(0) = 0 \):
\[ f(0) = \frac{1}{0^{-5} + 0^{-7} + 2} + C = 0 \]
Since \( 0^{-5} \) and \( 0^{-7} \) are undefined, we consider the limit as \( x \to 0 \):
\[ \lim_{x \to 0} f(x) = \frac{1}{2} + C = 0 \implies C = -\frac{1}{2} \]

### Step 8: Final expression for \( f(x) \)
Thus, we have:
\[ f(x) = \frac{1}{x^{-5} + x^{-7} + 2} - \frac{1}{2} \]

### Step 9: Calculate \( f(1) \)
Now, we find \( f(1) \):
\[ f(1) = \frac{1}{1^{-5} + 1^{-7} + 2} - \frac{1}{2} = \frac{1}{1 + 1 + 2} - \frac{1}{2} = \frac{1}{4} - \frac{1}{2} = \frac{1}{4} - \frac{2}{4} = -\frac{1}{4} \]

### Final Answer
The value of \( f(1) \) is:
\[ \boxed{\frac{1}{4}} \]