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.

Learn how to integrate the function 1/(x*(1 + x^2)) step-by-step using partial fraction decomposition and integration techniques. This problem explores advanced integration concepts and is suitable for advanced high school students.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation