Question

Evaluate the integral: \[ \int \frac{x}{x+2}\, dx \]

• A. \(x - 2\ln|x+2| + C \)
• B. \(x + 2\ln|x+2| + C \)
• C. \(x - \ln|x+2| + C \)
• D. \( \ln|x+2| + C \)

Hint:

For integrals of the form \( \frac{x}{x+a} \), rewrite the numerator as \(x+a-a\). This converts the expression into \(1 - \frac{a}{x+a}\), making the integration straightforward.

Answer 1

The Correct Option is A

Step 1: Understanding the Question:
The integrand is an improper rational fraction where the degree of the numerator is equal to the degree of the denominator.
Direct integration is not possible without simplifying the expression first.
Step 2: Key Formula or Approach:
1. Simplify the fraction by adding and subtracting terms in the numerator (algebraic manipulation).
2. Alternatively, use polynomial long division.
3. Use the basic integration rules: \( \int 1 dx = x + C \) and \( \int \frac{1}{u} du = \ln|u| + C \).
Step 3: Detailed Explanation:
Consider the integrand \( \frac{x}{x+2} \).
Modify the numerator to match the denominator:
\[ \frac{x}{x+2} = \frac{(x+2) - 2}{x+2} \]
Split the fraction into two separate terms:
\[ = \frac{x+2}{x+2} - \frac{2}{x+2} \]
\[ = 1 - \frac{2}{x+2} \]
Now, integrate the simplified expression:
\[ \int \left( 1 - \frac{2}{x+2} \right) dx \]
\[ = \int 1 dx - 2 \int \frac{1}{x+2} dx \]
Applying the standard integration results:
\[ = x - 2\ln|x+2| + C \]
Step 4: Final Answer:
The result of the integral is \( x - 2\ln|x+2| + C \).