Question

\( \int \frac{2x^2 + 4x + 3}{x^2 + x + 1} \, dx = \)

• A. \( 2\log_e|x^2+x+1| + C \)
• B. \( x\log_e|x^2+x+1| + C \)
• C. \( \frac{1}{2}\log_e|x^2+x+1| + C \)
• D. \( 2x + \log_e|x^2+x+1| + C \)
• E. \( x + 2\log_e|x^2+x+1| + C \)

Hint:

Always try division when numerator degree $\ge$ denominator degree.

Answer 1

The Correct Option is D

Step 1: Understanding the Concept:
The integral is of a rational function where the degree of the numerator is equal to the degree of the denominator. In such cases, the first step is to perform polynomial long division or algebraic manipulation to simplify the integrand into a polynomial part and a proper rational fraction.
Step 2: Key Formula or Approach:
1. Rewrite the numerator to contain a multiple of the denominator. 2. Split the fraction into simpler parts. 3. Integrate each part. The integral of the form \(\int \frac{f'(x)}{f(x)} dx\) is \(\ln|f(x)| + C\).
Step 3: Detailed Explanation:
The integrand is \(\frac{2x^2+4x+3}{x^2+x+1}\). We can rewrite the numerator to isolate a term that is a multiple of the denominator. We can see that \(2(x^2+x+1) = 2x^2 + 2x + 2\). Let's rewrite the numerator using this: \[ 2x^2+4x+3 = (2x^2 + 2x + 2) + 2x + 1 = 2(x^2+x+1) + (2x+1) \] Now, substitute this back into the integral: \[ \int \frac{2(x^2+x+1) + (2x+1)}{x^2+x+1} dx \] Split the integral into two parts: \[ \int \left( \frac{2(x^2+x+1)}{x^2+x+1} + \frac{2x+1}{x^2+x+1} \right) dx \] \[ \int 2 dx + \int \frac{2x+1}{x^2+x+1} dx \] The first integral is simple: \[ \int 2 dx = 2x \] For the second integral, notice that the numerator \(2x+1\) is the derivative of the denominator \(x^2+x+1\). This is an integral of the form \(\int \frac{f'(x)}{f(x)} dx\). \[ \int \frac{2x+1}{x^2+x+1} dx = \ln|x^2+x+1| \] (Since \(x^2+x+1\) has a discriminant of \(1^2 - 4(1)(1) = -3<0\), it is always positive, so the absolute value is not strictly necessary but is conventionally included). Combining the results and adding the constant of integration C: \[ 2x + \ln|x^2+x+1| + C \] Step 4: Final Answer:
The integral is \(2x + \log_e |x^2+x+1| + C\).