If the integral `I=int(tanx)/(5+7tan^(2)x)dx=kln |f(x)|+C` (where C is the integration constant) and `f(0)=5/7`, then the value of `f((pi)/(4))` is equal to

Answer

To solve the integral \( I = \int \frac{\tan x}{5 + 7 \tan^2 x} \, dx \) and find \( f\left(\frac{\pi}{4}\right) \) given that \( I = k \ln |f(x)| + C \) and \( f(0) = \frac{5}{7} \), we will follow these steps:

### Step 1: Substitute for \( t \)
Let \( t = 5 + 7 \tan^2 x \). Then, differentiate \( t \):
\[ dt = 14 \tan x \sec^2 x \, dx \Rightarrow dx = \frac{dt}{14 \tan x \sec^2 x} \]

### Step 2: Rewrite the integral
From the substitution, we have:
\[ \tan^2 x = \frac{t - 5}{7} \quad \text{and} \quad \sec^2 x = 1 + \tan^2 x = 1 + \frac{t - 5}{7} = \frac{t + 2}{7} \]
Thus, the integral becomes:
\[ I = \int \frac{\tan x}{t} \cdot \frac{dt}{14 \tan x \cdot \frac{t + 2}{7}} = \int \frac{7}{14} \cdot \frac{1}{t(t + 2)} \, dt = \frac{1}{2} \int \frac{1}{t(t + 2)} \, dt \]

### Step 3: Partial fraction decomposition
We can express:
\[ \frac{1}{t(t + 2)} = \frac{A}{t} + \frac{B}{t + 2} \]
Multiplying through by \( t(t + 2) \) gives:
\[ 1 = A(t + 2) + Bt \]
Setting \( t = 0 \) gives \( A = \frac{1}{2} \). Setting \( t = -2 \) gives \( B = -\frac{1}{2} \). Thus:
\[ \frac{1}{t(t + 2)} = \frac{1/2}{t} - \frac{1/2}{t + 2} \]

### Step 4: Integrate
Now we can integrate:
\[ I = \frac{1}{2} \left( \frac{1}{2} \ln |t| - \frac{1}{2} \ln |t + 2| \right) + C = \frac{1}{4} \ln \left| \frac{t}{t + 2} \right| + C \]

### Step 5: Substitute back for \( t \)
Substituting back \( t = 5 + 7 \tan^2 x \):
\[ I = \frac{1}{4} \ln \left| \frac{5 + 7 \tan^2 x}{7 + 7 \tan^2 x} \right| + C \]

### Step 6: Identify \( f(x) \)
From the equation \( I = k \ln |f(x)| + C \), we can identify:
\[ f(x) = \frac{5 + 7 \tan^2 x}{7 + 7 \tan^2 x} \]

### Step 7: Use the condition \( f(0) = \frac{5}{7} \)
At \( x = 0 \):
\[ f(0) = \frac{5 + 7 \tan^2(0)}{7 + 7 \tan^2(0)} = \frac{5}{7} \]
This condition is satisfied.

### Step 8: Calculate \( f\left(\frac{\pi}{4}\right) \)
Now, we find \( f\left(\frac{\pi}{4}\right) \):
\[ f\left(\frac{\pi}{4}\right) = \frac{5 + 7 \tan^2\left(\frac{\pi}{4}\right)}{7 + 7 \tan^2\left(\frac{\pi}{4}\right)} = \frac{5 + 7 \cdot 1}{7 + 7 \cdot 1} = \frac{12}{14} = \frac{6}{7} \]

### Final Answer
Thus, the value of \( f\left(\frac{\pi}{4}\right) \) is:
\[ \boxed{\frac{6}{7}} \]