Please integrate log(x^2 +1)/x. The answer will require the Li_2(x) function. Please proceed step by step.

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.

---

### 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.

Learn how to solve the integral ∫(log(x^2 + 1)/x) dx step by step, using substitution and the dilogarithm function Li_2(x). This detailed explanation covers advanced calculus concepts and special functions like the dilogarithm, suitable for undergraduate and advanced calculus students.

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