We are asked to evaluate the indefinite integral:

$\int \frac{dx}{x^2 - 2ax + 1}$

Step 1: Simplify the denominator

The expression in the denominator is a quadratic. We begin by completing the square on the quadratic expression $x^2 - 2ax + 1$:

$x^2 - 2ax + 1 = (x - a)^2$

Thus, the integral becomes:

$\int \frac{dx}{(x - a)^2}$

Step 2: Substitution

We now use a simple substitution to simplify the integral. Let:

$u = x - a \quad \text{so} \quad du = dx$

The integral then becomes:

$\int \frac{du}{u^2}$

Step 3: Solve the new integral

The integral of $\frac{1}{u^2}$ is:

$\int u^{-2} \, du = -\frac{1}{u}$

Thus, we get:

$-\frac{1}{u} + C = -\frac{1}{x - a} + C$

Final Answer:

The solution to the integral is:

$\int \frac{dx}{x^2 - 2ax + 1} = -\frac{1}{x - a} + C$

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Would you like more details or have any questions about the steps? Here are some related topics you might find interesting:

1. How to complete the square for quadratic expressions.
2. The process of substituting variables in integrals.
3. How to integrate rational functions involving quadratics.
4. The relationship between indefinite integrals and antiderivatives.
5. Solving integrals of the form $\int \frac{dx}{(x - a)^2}$.

Tip: Remember, the integral of $\frac{1}{u^2}$ is a basic result, and it's often useful when dealing with functions that involve squared terms in the denominator!