`int( dx )/( x ( x^(7) +1)) ` is equal to

Options

A. `log | ( x^(7))/( x^(7) +1)|+C`
B. `(1)/( 7 ) log | ( x^(7))/( x^(7) + 1) | +C`
C. `log | ( x^(7) + 1)/( x^(7)) | +C`
D. `(1)/( 7) log | (x^(7) + 1)/( x^(7)) |+ C`

Answer

To solve the integral \( \int \frac{dx}{x(x^7 + 1)} \), we can follow these steps:

### Step 1: Rewrite the Integral
Let
\[ I = \int \frac{dx}{x(x^7 + 1)} \]

### Step 2: Multiply by \( x^6 \)
We can multiply the numerator and denominator by \( x^6 \):
\[ I = \int \frac{x^6 \, dx}{x^7(x^7 + 1)} = \int \frac{dx}{x^7 + 1} \]

### Step 3: Substitution
Now, let \( t = x^7 \). Then, we differentiate:
\[ dt = 7x^6 \, dx \quad \Rightarrow \quad dx = \frac{dt}{7x^6} \]
Since \( x^6 = t^{6/7} \), we substitute:
\[ dx = \frac{dt}{7t^{6/7}} \]

### Step 4: Substitute in the Integral
Substituting \( t \) into the integral gives:
\[ I = \int \frac{1}{t(t + 1)} \cdot \frac{dt}{7t^{6/7}} = \frac{1}{7} \int \frac{dt}{t(t + 1)} \]

### Step 5: Partial Fraction Decomposition
Using partial fractions, we can write:
\[ \frac{1}{t(t + 1)} = \frac{1}{t} - \frac{1}{t + 1} \]
Thus,
\[ I = \frac{1}{7} \left( \int \frac{1}{t} \, dt - \int \frac{1}{t + 1} \, dt \right) \]

### Step 6: Integrate
Integrating each term:
\[ I = \frac{1}{7} \left( \log |t| - \log |t + 1| \right) + C \]

### Step 7: Substitute Back
Substituting back \( t = x^7 \):
\[ I = \frac{1}{7} \left( \log |x^7| - \log |x^7 + 1| \right) + C \]

### Step 8: Simplify
Using the properties of logarithms:
\[ I = \frac{1}{7} \log \left| \frac{x^7}{x^7 + 1} \right| + C \]

### Final Result
Thus, the final result for the integral is:
\[ \int \frac{dx}{x(x^7 + 1)} = \frac{1}{7} \log \left| \frac{x^7}{x^7 + 1} \right| + C \]