To solve the integral of $\frac{\log(x^2 + 1)}{x} \, dx$, we proceed step by step and use special functions like the dilogarithm $\text{Li}_2(x)$ when necessary. Here's how to tackle this problem:

Step 1: Recognize the structure of the integrand

We need to evaluate: $I = \int \frac{\log(x^2 + 1)}{x} \, dx$ This integral is non-trivial and involves logarithmic and rational functions. To approach it, we can use substitution and the polylogarithmic functions, including the dilogarithm function $\text{Li}_2(x)$.

Step 2: Substitution

Let's try the substitution: $u = x^2 + 1$ Then, we compute the differential of $u$: $du = 2x \, dx$ Thus, $x \, dx = \frac{du}{2}$ Rewriting the integral in terms of $u$, we get: $I = \frac{1}{2} \int \frac{\log(u)}{u - 1} \, du$

Step 3: Express in terms of the dilogarithm function $\text{Li}_2(x)$

The integral $\int \frac{\log(u)}{u - 1} \, du$ is known to be related to the dilogarithm function $\text{Li}_2(x)$. Specifically, we can use the following known result from integral tables or advanced calculus: $\int \frac{\log(u)}{u - 1} \, du = \text{Li}_2(1 - u) + \log(u) \log(u - 1)$

Step 4: Substitute back $u = x^2 + 1$

Now that we have the result of the integral in terms of $u$, we substitute back $u = x^2 + 1$: $I = \frac{1}{2} \left( \text{Li}_2(1 - (x^2 + 1)) + \log(x^2 + 1) \log(x^2) \right)$ Simplifying: $I = \frac{1}{2} \left( \text{Li}_2(-x^2) + \log(x^2 + 1) \cdot 2 \log(x) \right)$ $I = \frac{1}{2} \text{Li}_2(-x^2) + \log(x) \log(x^2 + 1)$

Step 5: Final answer

The final expression for the integral is: $I = \frac{1}{2} \text{Li}_2(-x^2) + 2 \log(x) \log(x^2 + 1) + C$ where $C$ is the constant of integration.

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Would you like further details on any step? Here are related questions for additional practice:

1. How can you apply the substitution method to simplify integrals involving logarithmic functions?
2. What are some other integrals where the dilogarithm $\text{Li}_2(x)$ function appears?
3. How is the dilogarithm function defined and what are its properties?
4. What other special functions, besides $\text{Li}_2(x)$, arise in integral calculus?
5. Can you compute the definite integral of $\frac{\log(x^2 + 1)}{x}$ from $x = 1$ to $x = 2$?

Tip: Familiarize yourself with special functions like the dilogarithm function $\text{Li}_2(x)$ as they often arise in advanced calculus and mathematical physics.