To solve the integral of \( \frac{2x}{x^2 + 9} \), we'll proceed with the following steps:

\[
\int \frac{2x}{x^2 + 9} \, dx
\]

### Step 1: Recognize the structure of the integrand.
The numerator \( 2x \) is the derivative of the denominator \( x^2 + 9 \). This suggests that a simple substitution can be applied.

### Step 2: Apply substitution.
Let:
\[
u = x^2 + 9
\]
Then:
\[
du = 2x \, dx
\]

### Step 3: Rewrite the integral.
Now substitute into the integral:
\[
\int \frac{2x \, dx}{x^2 + 9} = \int \frac{du}{u}
\]

### Step 4: Integrate.
The integral of \( \frac{1}{u} \) is a standard result:
\[
\int \frac{du}{u} = \ln|u| + C
\]

### Step 5: Substitute back for \( u \).
Now, substitute \( u = x^2 + 9 \) back into the result:
\[
\ln|x^2 + 9| + C
\]

Since \( x^2 + 9 \) is always positive, we can drop the absolute value:
\[
\ln(x^2 + 9) + C
\]

### Final Answer:
\[
\int \frac{2x}{x^2 + 9} \, dx = \ln(x^2 + 9) + C
\]

Would you like more details or have any questions? Here are some related questions you might explore:

1. What if the numerator was \( 3x \) instead of \( 2x \)?
2. How would the integral change if the denominator were \( x^2 + 16 \)?
3. What are the steps for solving \( \int \frac{x}{x^2 + 9} \, dx \)?
4. How would you solve \( \int \frac{1}{x^2 + 9} \, dx \)?
5. How do we apply substitution in integrals in general?

**Tip**: Recognizing when the numerator is a derivative of the denominator is a helpful shortcut in integration!

Learn how to solve the integral of 2x/(x^2 + 9) using substitution and logarithmic integration. Follow a detailed step-by-step solution that covers important calculus concepts, including substitution and logarithmic integrals. Suitable for college-level 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