Question

If \[ \int \frac{2x+1}{x^2+x+1}\,dx \] is equal to:

• A. \(\ln(x^2+x+1)+C\)
• B. \(\frac12\ln(x^2+x+1)+C\)
• C. \(\tan^{-1}(x)+C\)
• D. \(\frac{x}{x^2+x+1}+C\)

Hint:

Look for the pattern \(\frac{f'(x)}{f(x)}\). It immediately gives a logarithmic integral.

Answer 1

The Correct Option is A

Concept: Whenever the numerator is the derivative of the denominator, use the standard formula: \[ \int \frac{f'(x)}{f(x)}\,dx = \ln|f(x)|+C \]

Step 1: Identify the denominator \[ f(x)=x^2+x+1 \] Differentiate: \[ f'(x)=2x+1 \]

Step 2: Compare numerator and derivative Numerator: \[ 2x+1 \] Derivative of denominator: \[ 2x+1 \] They are exactly the same.

Step 3: Apply the standard result \[ \int \frac{2x+1}{x^2+x+1}\,dx = \ln|x^2+x+1|+C \] Since \[ x^2+x+1>0 \] for all real \(x\), \[ =\ln(x^2+x+1)+C \] Final Answer: \[ \boxed{\ln(x^2+x+1)+C} \]