`int (1)/(x(x^(3)-1))dx`

Answer

To solve the integral \( \int \frac{1}{x(x^3 - 1)} \, dx \), we can follow these steps:

### Step 1: Rewrite the Integral
We start by rewriting the integral:
\[ I = \int \frac{1}{x(x^3 - 1)} \, dx \]

### Step 2: Factor the Denominator
Next, we factor the denominator:
\[ x^3 - 1 = (x - 1)(x^2 + x + 1) \]
Thus, we can rewrite the integral as:
\[ I = \int \frac{1}{x (x - 1)(x^2 + x + 1)} \, dx \]

### Step 3: Partial Fraction Decomposition
We will use partial fraction decomposition to break down the integrand:
\[ \frac{1}{x (x - 1)(x^2 + x + 1)} = \frac{A}{x} + \frac{B}{x - 1} + \frac{Cx + D}{x^2 + x + 1} \]
Multiplying through by the denominator \( x (x - 1)(x^2 + x + 1) \) gives:
\[ 1 = A(x - 1)(x^2 + x + 1) + Bx(x^2 + x + 1) + (Cx + D)x(x - 1) \]

### Step 4: Solve for Coefficients
To find the coefficients \( A, B, C, D \), we can substitute suitable values for \( x \) or equate coefficients from both sides. After solving, we find:
- \( A = \frac{1}{3} \)
- \( B = \frac{1}{3} \)
- \( C = 0 \)
- \( D = -\frac{1}{3} \)

Thus, we can rewrite the integral as:
\[ I = \int \left( \frac{1/3}{x} + \frac{1/3}{x - 1} - \frac{1/3}{x^2 + x + 1} \right) \, dx \]

### Step 5: Integrate Each Term
Now we can integrate each term separately:
\[ I = \frac{1}{3} \int \frac{1}{x} \, dx + \frac{1}{3} \int \frac{1}{x - 1} \, dx - \frac{1}{3} \int \frac{1}{x^2 + x + 1} \, dx \]

The first two integrals yield:
\[ \frac{1}{3} \ln |x| + \frac{1}{3} \ln |x - 1| \]

For the third integral, we complete the square in the denominator:
\[ x^2 + x + 1 = \left(x + \frac{1}{2}\right)^2 + \frac{3}{4} \]
Thus,
\[ \int \frac{1}{x^2 + x + 1} \, dx = \frac{2}{\sqrt{3}} \tan^{-1} \left(\frac{2x + 1}{\sqrt{3}}\right) \]

### Step 6: Combine Results
Combining all parts, we have:
\[ I = \frac{1}{3} \ln |x| + \frac{1}{3} \ln |x - 1| - \frac{1}{3} \cdot \frac{2}{\sqrt{3}} \tan^{-1} \left(\frac{2x + 1}{\sqrt{3}}\right) + C \]

### Final Answer
Thus, the final answer is:
\[ I = \frac{1}{3} \ln |x(x - 1)| - \frac{2}{3\sqrt{3}} \tan^{-1} \left(\frac{2x + 1}{\sqrt{3}}\right) + C \]

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