Question

\(\int \frac{x}{x+2}dx\) is equal to

• A. \(x+2log(x+2)+c\)
• B. \(x+log(x+2)+c\)
• C. \(x-log(x+2)+c\)
• D. \(x-2log(x+2)+c\)

Hint:

Write x as (x+2) - 2 so the fraction splits into 1 minus 2/(x+2).

Answer 1

The Correct Option is D

Step 1: Substitute:
Let $t=x+2$, so $x=t-2$ and $dx=dt$.

Step 2: Integrate in t:
$\int\frac{t-2}{t}dt=\int\left(1-\frac2t\right)dt=t-2\log t+c_1$.

Step 3: Go back to x:
$t-2\log t=x+2-2\log(x+2)$. The constant 2 merges with $c_1$, so the result is $x-2\log(x+2)+c$.

Step 4: Check by differentiating:
$\frac{d}{dx}\left[x-2\log(x+2)\right]=1-\frac{2}{x+2}=\frac{x}{x+2}$. Correct.

Final Answer:
The answer is $x-2\log(x+2)+c$ (option D). \[ \boxed{x-2\log(x+2)+c} \]