Evaluate $ \frac{2x+1}{x^2+x+2}\, dx$

1. $log(2x+1) + c$ where c is an arbitrary constant

2. $log\!\left(\frac{2x+1} {x^2+x+2}\right) + c$ where c is an arbitrary constant

3. $log(x^2+x+2) + c$  where c is an arbitrary constant

4.  $log\!\left(\tfrac{1}{2}\right) + c$ where c is an arbitrary constant

Step 1: Let the denominator be $$f(x) = x^2 + x + 2.$$ Then, $$f'(x) = 2x + 1,$$ which is the numerator.

Step 2: Apply the rule: $$\int \frac{f'(x)}{f(x)} \, dx = \ln|f(x)| + C.$$

Step 3: Therefore, $$\int \frac{2x+1}{x^2+x+2}\, dx = \ln|x^2+x+2| + C.$$

Final Answer: $\log(x^2+x+2) + c$

Correct Option: 3