Question

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

Hint:

For \(\int \frac{ax+b}{cx+d} dx\), the result is always \(\frac{a}{c}x + (\dots)\log|cx+d|\). This "linear over linear" form can be solved in seconds by looking at the coefficients.

Answer 1

Step 1: Understanding the Concept:
When the degree of the numerator is greater than or equal to the degree of the denominator, we use algebraic manipulation or division.

Step 2: Key Formula or Approach:
1. Addition and subtraction in the numerator: \(\frac{x}{x+2} = \frac{x+2-2}{x+2}\).
2. \(\int \frac{1}{x+a} dx = \log|x+a| + C\).

Step 3: Detailed Explanation:
Rewrite the expression inside the integral:
\[ \frac{x}{x+2} = \frac{(x+2) - 2}{x+2} = \frac{x+2}{x+2} - \frac{2}{x+2} = 1 - \frac{2}{x+2} \] Integrate:
\[ \int \left(1 - \frac{2}{x+2}\right) dx = \int 1 dx - 2 \int \frac{1}{x+2} dx \] \[ = x - 2\log|x+2| + C \]
Step 4: Final Answer:
The integral is \(x - 2\log|x + 2| + C\).