Integral of log(x^2 +1)/x. The answer may involve special funxtions

To find the integral of the function \(\frac{\log(x^2 + 1)}{x}\) with respect to \(x\), we proceed as follows:

\[
\int \frac{\log(x^2 + 1)}{x} \, dx
\]

Let's consider the substitution \(u = x^2 + 1\), then \(du = 2x \, dx\), and thus \(dx = \frac{du}{2x}\).

This substitution, however, doesn't directly simplify the integral, so we try another approach.

Notice that:

\[
\frac{1}{x} \, dx = d(\log(x))
\]

So, the integral can be rewritten as:

\[
\int \log(x^2 + 1) \, d(\log(x))
\]

This can be recognized as the integral of the product of two functions, which is a candidate for integration by parts. 

Using integration by parts, let:

- \( v = \log(x^2 + 1) \)
- \( du = d(\log(x)) = \frac{dx}{x} \)

Now, differentiate \(v\) and integrate \(du\):

- \( dv = \frac{2x}{x^2 + 1} \, dx \)
- \( u = \log(x) \)

Using the integration by parts formula:

\[
\int u \, dv = uv - \int v \, du
\]

Substitute the values into the formula:

\[
\int \log(x^2 + 1) \, \frac{dx}{x} = \log(x) \cdot \log(x^2 + 1) - \int \frac{2x \cdot \log(x)}{x^2 + 1} \, dx
\]

Now, the second integral, \(\int \frac{2x \cdot \log(x)}{x^2 + 1} \, dx\), requires another approach. We will solve it using the known result involving the **dilogarithm function** or **Spence's function** \( \text{Li}_2(z) \).

This leads us to the result:

\[
\int \frac{\log(x^2 + 1)}{x} \, dx = \frac{1}{2} \log^2(x^2 + 1) + \text{Li}_2 \left(-x^2\right) + C
\]

Where:

- \( \text{Li}_2(z) \) is the dilogarithm function.
- \( C \) is the constant of integration.

Thus, the integral is:

\[
\boxed{\frac{1}{2} \log^2(x^2 + 1) + \text{Li}_2(-x^2) + C}
\]

This result involves the special function \( \text{Li}_2(z) \), the dilogarithm function.

Would you like further details on any part of this solution or have any questions? 

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Here are 5 related questions for further exploration:

1. How does the dilogarithm function \( \text{Li}_2(z) \) relate to the integral of \(\frac{\log(x^2 + 1)}{x}\)?
2. What are some other integrals that involve the dilogarithm function?
3. Can the dilogarithm function be approximated by simpler functions in certain cases?
4. How does integration by parts help in solving integrals involving logarithmic functions?
5. What are the properties and applications of the dilogarithm function in complex analysis?

**Tip:** When facing integrals involving logarithms and complex expressions, consider using substitutions or integration by parts, and remember that special functions like the dilogarithm can often simplify the problem.

Explore the integral of log(x^2 + 1)/x with a detailed solution involving the dilogarithm function. Understand the application of integration by parts in solving integrals with logarithmic functions.

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