To solve the integral \( \int \frac{x}{x^4 - 1} \, dx \), we can follow these steps:
### Step 1: Rewrite the Integral
We start with the integral:
\[
I = \int \frac{x}{x^4 - 1} \, dx
\]
We can factor the denominator:
\[
x^4 - 1 = (x^2 - 1)(x^2 + 1) = (x - 1)(x + 1)(x^2 + 1)
\]
Thus, we rewrite the integral as:
\[
I = \int \frac{x}{(x - 1)(x + 1)(x^2 + 1)} \, dx
\]
### Step 2: Use Substitution
We will use the substitution \( t = x^2 \). Then, we have:
\[
dt = 2x \, dx \quad \Rightarrow \quad x \, dx = \frac{dt}{2}
\]
Substituting into the integral gives:
\[
I = \int \frac{1}{(t - 1)(t + 1)(t + 1)} \cdot \frac{dt}{2}
\]
This simplifies to:
\[
I = \frac{1}{2} \int \frac{1}{(t - 1)(t + 1)(t + 1)} \, dt
\]
### Step 3: Partial Fraction Decomposition
Next, we perform partial fraction decomposition on:
\[
\frac{1}{(t - 1)(t + 1)(t + 1)} = \frac{A}{t - 1} + \frac{B}{t + 1} + \frac{C}{(t + 1)^2}
\]
Multiplying through by the denominator \( (t - 1)(t + 1)^2 \) gives:
\[
1 = A(t + 1)^2 + B(t - 1)(t + 1) + C(t - 1)
\]
Expanding and solving for \( A, B, C \) leads to a system of equations.
### Step 4: Solve for Coefficients
By substituting suitable values for \( t \) (like \( t = 1 \), \( t = -1 \), etc.), we can solve for \( A, B, C \).
### Step 5: Integrate Each Term
After finding \( A, B, C \), we can integrate each term separately:
\[
\int \frac{A}{t - 1} \, dt, \quad \int \frac{B}{t + 1} \, dt, \quad \int \frac{C}{(t + 1)^2} \, dt
\]
### Step 6: Substitute Back
After integrating, substitute back \( t = x^2 \) to express the result in terms of \( x \).
### Final Result
The final result of the integral will be of the form:
\[
\frac{1}{4} \log\left|\frac{x^2 - 1}{x^2 + 1}\right| + C
\]
where \( C \) is the constant of integration.
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