To solve the integral

$\int \frac{1}{x \cdot (1 + x^2)} \, dx,$

we can use the method of partial fraction decomposition. First, let's rewrite the integrand in a more convenient form. We decompose the fraction as follows:

$\frac{1}{x(1 + x^2)} = \frac{A}{x} + \frac{Bx + C}{1 + x^2}.$

Now we need to find the constants $A$, $B$, and $C$. To do this, we write:

$1 = A(1 + x^2) + (Bx + C)x.$

Expanding the right-hand side, we get:

$1 = A + Ax^2 + Bx^2 + Cx.$

Grouping like terms, we have:

$1 = A + (A + B)x^2 + Cx.$

To satisfy this equation for all $x$, the coefficients of $x^2$, $x$, and the constant term must match on both sides. This gives us the following system of equations:

$A = 1, \quad C = 0, \quad \text{and} \quad A + B = 0.$

From $A + B = 0$, we get $B = -A$. Since $A = 1$, we have $B = -1$. Therefore, the partial fraction decomposition is:

$\frac{1}{x(1 + x^2)} = \frac{1}{x} - \frac{x}{1 + x^2}.$

Now we can integrate each term separately:

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

The integral of $\frac{1}{x}$ with respect to $x$ is:

$\int \frac{1}{x} \, dx = \ln|x|.$

For the integral of $\frac{x}{1 + x^2}$ with respect to $x$, we use the substitution $u = 1 + x^2$, $du = 2x \, dx$. Therefore:

$\int \frac{x}{1 + x^2} \, dx = \frac{1}{2} \int \frac{2x}{1 + x^2} \, dx = \frac{1}{2} \int \frac{du}{u} = \frac{1}{2} \ln|u| = \frac{1}{2} \ln|1 + x^2|.$

Putting these results together, we get:

$\int \frac{1}{x(1 + x^2)} \, dx = \ln|x| - \frac{1}{2} \ln|1 + x^2| + C.$

Therefore, the solution to the integral is:

$\boxed{\ln|x| - \frac{1}{2} \ln|1 + x^2| + C}.$

Would you like any details or have any questions?

Here are 5 numbered questions you can ask:

1. How do you integrate $x \cdot e^x$?
2. Can you explain the method of integration by parts?
3. How do you solve a definite integral from 0 to 1 of $x^2$?
4. What is the integral of $\sin(x) \cdot \cos(x)$?
5. How do you find the area under a curve using integration?

Tip: Partial fraction decomposition is a useful technique for integrating rational functions. It involves expressing the integrand as a sum of simpler fractions that can be integrated separately.