If `int((1-x^(7)))/(x(1+x^(7)))dx=Plog|x|+Qlog|x^(7)+1|+c`, then

Options

A. `7P+2Q=1`
B. `7P-2Q=1`
C. `7P+2Q=0`
D. none of these

Answer

To solve the integral

\[ I = \int \frac{1 - x^7}{x(1 + x^7)} \, dx, \]

we can break it down into simpler parts.

### Step 1: Rewrite the Integral

We can express the integrand as follows:

\[ I = \int \left( \frac{1 + x^7 - 2x^7}{x(1 + x^7)} \right) \, dx = \int \left( \frac{1 + x^7}{x(1 + x^7)} - \frac{2x^7}{x(1 + x^7)} \right) \, dx. \]

This simplifies to:

\[ I = \int \frac{1}{x} \, dx - 2 \int \frac{x^6}{1 + x^7} \, dx. \]

### Step 2: Solve the First Integral

The first integral is straightforward:

\[ \int \frac{1}{x} \, dx = \log |x| + C_1. \]

### Step 3: Solve the Second Integral

For the second integral, we can use a substitution. Let:

\[ t = 1 + x^7 \implies dt = 7x^6 \, dx \implies dx = \frac{dt}{7x^6}. \]

Now, substituting \(x^6 = (t - 1)^{6/7}\):

\[ \int \frac{x^6}{1 + x^7} \, dx = \int \frac{x^6}{t} \cdot \frac{dt}{7x^6} = \frac{1}{7} \int \frac{1}{t} \, dt = \frac{1}{7} \log |t| + C_2 = \frac{1}{7} \log |1 + x^7| + C_2. \]

### Step 4: Combine the Results

Putting it all together, we have:

\[ I = \log |x| - 2 \cdot \frac{1}{7} \log |1 + x^7| + C. \]

This simplifies to:

\[ I = \log |x| - \frac{2}{7} \log |1 + x^7| + C. \]

### Step 5: Compare with the Given Form

The problem states that:

\[ I = P \log |x| + Q \log |1 + x^7| + C. \]

From our result, we can identify:

- \(P = 1\)
- \(Q = -\frac{2}{7}\)

### Step 6: Analyze the Options

Now, we need to check the options given in the problem. The expression \(7P + 2Q\) becomes:

\[ 7(1) + 2\left(-\frac{2}{7}\right) = 7 - \frac{4}{7} = \frac{49}{7} - \frac{4}{7} = \frac{45}{7}. \]

Since \(7P + 2Q \neq 1\), we conclude that the correct option is that none of the provided options are correct.