https://file.aiquickdraw.com/mathbot/file/1725454926797jv1snupp.jpg

The integral to evaluate is:

\[
\int \frac{1 - 4x + x^2}{2x} \, dx
\]

### Solution

1. **Simplify the integrand:**
   \[
   \frac{1 - 4x + x^2}{2x} = \frac{1}{2x} - 2 + \frac{x^2}{2x}
   \]
   Simplifying further:
   \[
   = \frac{1}{2x} - 2 + \frac{x}{2}
   \]

2. **Separate the integral:**
   \[
   \int \left(\frac{1}{2x} - 2 + \frac{x}{2}\right) dx = \int \frac{1}{2x} dx - \int 2 \, dx + \int \frac{x}{2} dx
   \]

3. **Integrate each term separately:**

   - For \(\int \frac{1}{2x} dx\):
     \[
     \int \frac{1}{2x} dx = \frac{1}{2} \ln|x|
     \]

   - For \(\int 2 \, dx\):
     \[
     \int 2 \, dx = 2x
     \]

   - For \(\int \frac{x}{2} dx\):
     \[
     \int \frac{x}{2} dx = \frac{1}{2} \cdot \frac{x^2}{2} = \frac{x^2}{4}
     \]

4. **Combine the results:**
   \[
   \frac{1}{2} \ln|x| - 2x + \frac{x^2}{4} + C
   \]
   where \( C \) is the constant of integration.

### Final Answer:
\[
\int \frac{1 - 4x + x^2}{2x} \, dx = \frac{1}{2} \ln|x| - 2x + \frac{x^2}{4} + C
\]

Would you like any more details or have any questions about this problem?

Here are 5 related questions for further exploration:

1. How would you evaluate the integral of \(\frac{x^2 + 3x + 2}{x}\)?
2. What is the result of integrating \(\frac{1}{x}\) directly?
3. How do you approach integrals that involve logarithmic functions?
4. Can this integral be solved using substitution? If so, how?
5. How does the method of partial fractions apply to integrals similar to this one?

**Tip:** Always try to simplify the integrand as much as possible before integrating. This often makes the integration process more straightforward.

Evaluate the integral \( \int \frac{1 - 4x + x^2}{2x} \, dx \)

Learn how to evaluate the integral \( \int \frac{1 - 4x + x^2}{2x} \, dx \) with a detailed step-by-step solution. This problem covers key calculus concepts and algebraic manipulation techniques. Suitable for undergraduate calculus 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