To integrate the expression \(\int \frac{x}{(x+1)^2} \, dx\), we can use substitution and some simplification.

### Solution:
1. **Substitute** \(u = x + 1\), which implies that \(du = dx\) and \(x = u - 1\).

   This substitution changes the integral as follows:
   \[
   \int \frac{x}{(x+1)^2} \, dx = \int \frac{u - 1}{u^2} \, du
   \]

2. **Simplify** the integrand:
   \[
   = \int \left( \frac{u}{u^2} - \frac{1}{u^2} \right) \, du = \int \left( \frac{1}{u} - \frac{1}{u^2} \right) \, du
   \]

3. **Separate the integrals**:
   \[
   = \int \frac{1}{u} \, du - \int \frac{1}{u^2} \, du
   \]

4. **Integrate each term**:
   - For \(\int \frac{1}{u} \, du\), the result is \(\ln |u|\).
   - For \(\int \frac{1}{u^2} \, du\), rewrite it as \(u^{-2}\), and integrate to get \(-\frac{1}{u}\).

   So,
   \[
   \int \frac{1}{u} \, du - \int \frac{1}{u^2} \, du = \ln |u| + \frac{1}{u} + C
   \]

5. **Substitute back** \(u = x + 1\):
   \[
   = \ln |x + 1| + \frac{1}{x + 1} + C
   \]

### Final Answer:
\[
\int \frac{x}{(x+1)^2} \, dx = \ln |x + 1| + \frac{1}{x + 1} + C
\]

Would you like more details on any step, or have any questions?

---

### Related Questions:

1. How would you integrate \(\int \frac{x^2}{(x+1)^3} \, dx\)?
2. Can you use a different method to integrate \(\int \frac{x}{(x+1)^2} \, dx\)?
3. What happens if we have \(\int \frac{1}{(x+1)^2} \, dx\)?
4. How do you find the antiderivative of \(\int \frac{x}{(x+1)^n} \, dx\) for an arbitrary \(n\)?
5. What is the integral of \(\int \frac{x+2}{(x+1)^2} \, dx\)?

### Tip:
When integrating rational functions, substitution is often helpful, especially when the derivative of the denominator closely resembles the numerator.

Explore the detailed step-by-step solution for the integral of x/(x+1)^2. This problem utilizes substitution and simplification techniques, and covers key calculus concepts. Suitable for students in advanced high school or undergraduate-level calculus courses.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation