Question

\[ \int \frac{2x+1}{x^{2}+x+2}\, dx \ \text{ is} \]

• A. $\log(2x+1)+c$, where $c$ is an arbitrary constant
• B. $\log\!\left(\dfrac{2x+1}{x^{2}+x+2}\right)+c$, where $c$ is an arbitrary constant
• C. $\log(x^{2}+x+2)+c$, where $c$ is an arbitrary constant
• D. $\log\!\left(\tfrac{1}{2}\right)+c$, where $c$ is an arbitrary constant

Hint:

When numerator is the derivative of denominator, the integral is simply $\ln(\text{denominator})+c$.

Answer 1

The Correct Option is C

Step 1: Identify the structure.
We observe that the denominator is $x^{2}+x+2$. Its derivative is $(2x+1)$, which matches the numerator.

Step 2: Apply substitution.
Let $t = x^{2}+x+2 \ $\Rightarrow$ \ dt = (2x+1)\, dx$.

Step 3: Rewrite integral.
\[ \int \frac{2x+1}{x^{2}+x+2}\, dx = \int \frac{dt}{t} \]

Step 4: Evaluate.
\[ \int \frac{dt}{t} = \log|t|+c = \log(x^{2}+x+2)+c \]

Step 5: Conclusion.
The required result is $\log(x^{2}+x+2)+c$.