Find the indefinite integral. (Remember to use absolute values where appropriate. Use C for the constant of integration.)

x2 + 10x + 4
x3 + 15x2 + 12x + 1

dx

To find the indefinite integral of the given function, we need to integrate the expression with respect to \(x\). The function to integrate is:

\[
\frac{x^2 + 10x + 4}{x^3 + 15x^2 + 12x + 1} \, dx
\]

This type of integral is generally approached using the method of partial fraction decomposition or substitution. However, in this case, neither approach simplifies the integral straightforwardly. Let's attempt a substitution:

1. **Substitution:**

   Let \( u = x^3 + 15x^2 + 12x + 1 \).

   Then, the derivative \( du \) with respect to \( x \) is:

   \[
   du = (3x^2 + 30x + 12) \, dx
   \]

   Notice that the numerator \( x^2 + 10x + 4 \) is not exactly \( \frac{1}{3}(3x^2 + 30x + 12) \), but it can be rewritten to aid integration.

   Now, let's rewrite the integral:

   \[
   \int \frac{x^2 + 10x + 4}{x^3 + 15x^2 + 12x + 1} \, dx
   \]

   Multiply the numerator and denominator by 3:

   \[
   \int \frac{3(x^2 + 10x + 4)}{3(x^3 + 15x^2 + 12x + 1)} \, dx = \frac{1}{3} \int \frac{3x^2 + 30x + 12}{x^3 + 15x^2 + 12x + 1} \, dx
   \]

   Notice that the numerator \( 3x^2 + 30x + 12 \) is the derivative of \( u \), so this simplifies the integral to:

   \[
   \frac{1}{3} \int \frac{du}{u} = \frac{1}{3} \ln|u| + C
   \]

   Substituting back \( u = x^3 + 15x^2 + 12x + 1 \), we get:

   \[
   \frac{1}{3} \ln|x^3 + 15x^2 + 12x + 1| + C
   \]

So, the indefinite integral is:

\[
\int \frac{x^2 + 10x + 4}{x^3 + 15x^2 + 12x + 1} \, dx = \frac{1}{3} \ln|x^3 + 15x^2 + 12x + 1| + C
\]

Would you like any details or further explanation?

### Relative Questions
1. How do you perform substitution in integration?
2. What is the purpose of partial fraction decomposition?
3. How do you identify when to use logarithmic integration?
4. What is the importance of the constant of integration, \(C\)?
5. How do you differentiate a logarithmic function?
6. Can all rational functions be integrated using partial fractions?
7. How do you recognize if an integral requires a substitution?
8. What is the method for integrating trigonometric functions?

### Tip
When integrating a rational function, always consider whether the numerator is the derivative of the denominator; this often simplifies the process.

Learn how to find the indefinite integral of (x^2 + 10x + 4)/(x^3 + 15x^2 + 12x + 1) with detailed step-by-step solutions. This problem involves integration techniques such as partial fraction decomposition and substitution, along with logarithmic integration for rational functions. Suitable for college-level 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