Evaluate :
`int(dx)/(x(x^(7)+1))`

Answer

To evaluate the integral

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

we can follow these steps:

### Step 1: Rewrite the Integral
We start with the integral:

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

### Step 2: Multiply and Divide by \(x^6\)
To facilitate the integration, we multiply and divide by \(x^6\):

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

### Step 3: Use Substitution
Let \(t = x^7\). Then, the derivative \(dt = 7x^6 \, dx\) or \(dx = \frac{dt}{7x^6}\).

Substituting these into the integral gives:

\[ I = \int \frac{1}{7} \cdot \frac{dt}{t(t + 1)}. \]

### Step 4: Partial Fraction Decomposition
Next, we can decompose \(\frac{1}{t(t + 1)}\) into partial fractions:

\[ \frac{1}{t(t + 1)} = \frac{A}{t} + \frac{B}{t + 1}. \]

Multiplying through by \(t(t + 1)\) gives:

\[ 1 = A(t + 1) + Bt. \]

Setting \(t = 0\) gives \(A = 1\). Setting \(t = -1\) gives \(B = -1\). Thus, we have:

\[ \frac{1}{t(t + 1)} = \frac{1}{t} - \frac{1}{t + 1}. \]

### Step 5: Substitute Back into the Integral
Now substituting back into the integral:

\[ I = \frac{1}{7} \int \left( \frac{1}{t} - \frac{1}{t + 1} \right) dt. \]

### Step 6: Integrate
Integrating term by term:

\[ I = \frac{1}{7} \left( \ln |t| - \ln |t + 1| \right) + C. \]

### Step 7: Combine Logarithms
Using the property of logarithms:

\[ I = \frac{1}{7} \ln \left| \frac{t}{t + 1} \right| + C. \]

### Step 8: Substitute Back for \(t\)
Recall that \(t = x^7\):

\[ I = \frac{1}{7} \ln \left| \frac{x^7}{x^7 + 1} \right| + C. \]

### Final Answer
Thus, the final result for the integral is:

\[ \int \frac{dx}{x(x^7 + 1)} = \frac{1}{7} \ln \left| \frac{x^7}{x^7 + 1} \right| + C. \]