Question

Evaluate the integral: \( \displaystyle \int \frac{x}{x+2}\,dx \)

• A. \( x - 2\log|x+2| + C \)
• B. \( x + 2\log|x+2| + C \)
• C. \( \log|x+2| + C \)
• D. \( \frac{x^2}{2(x+2)} + C \)

Hint:

When integrating rational functions, try rewriting the numerator in terms of the denominator. This often converts the integral into a sum of simple standard integrals.

Answer 1

The Correct Option is A

Step 1: Understanding the Question:
The objective is to find the indefinite integral of a rational function where the degree of the numerator is equal to the degree of the denominator.
Step 2: Key Formula or Approach:
For such rational functions, we can simplify the expression by performing algebraic manipulation in the numerator to match the denominator.
We use the standard integral formula:
\[ \int \frac{1}{x+a} dx = \log|x+a| + C \] Step 3: Detailed Explanation:
We rewrite the numerator \(x\) as \((x + 2) - 2\) to facilitate division by the denominator.
\[ I = \int \frac{(x+2) - 2}{x+2} dx \] Distributing the denominator:
\[ I = \int \left( \frac{x+2}{x+2} - \frac{2}{x+2} \right) dx \] \[ I = \int \left( 1 - \frac{2}{x+2} \right) dx \] Integrating the terms separately:
\[ I = \int 1 dx - 2 \int \frac{1}{x+2} dx \] \[ I = x - 2\log|x+2| + C \] Step 4: Final Answer:
The value of the integral is \( x - 2\log|x+2| + C \).