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Evaluate the integral int(2x+1)/(x+2)dx. Choose 1 answer: (A) 2x-3ln |x+2|+C (B) ln |x+2|+C (C) x+ln |2x+1|+C (D) 2x-ln |x+2|+C

Evaluate the integral ∫2x+1x+2dx \int \frac{2 x+1}{x+2} d x .\newlineChoose 11 answer:\newline(A) 2x−3ln⁡∣x+2∣+C 2 x-3 \ln |x+2|+C \newline(B) ln⁡∣x+2∣+C \ln |x+2|+C \newline(C) x+ln⁡∣2x+1∣+C x+\ln |2 x+1|+C \newline(D) 2x−ln⁡∣x+2∣+C 2 x-\ln |x+2|+C

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Q. Evaluate the integral ∫2x+1x+2dx \int \frac{2 x+1}{x+2} d x .\newlineChoose 11 answer:\newline(A) 2x−3ln⁡∣x+2∣+C 2 x-3 \ln |x+2|+C \newline(B) ln⁡∣x+2∣+C \ln |x+2|+C \newline(C) x+ln⁡∣2x+1∣+C x+\ln |2 x+1|+C \newline(D) 2x−ln⁡∣x+2∣+C 2 x-\ln |x+2|+C
  1. Divide and Simplify: Let's do long division first to simplify the integral.\newline(2x+1)/(x+2)(2x+1)/(x+2) can be divided to get 22 with a remainder of −3-3.\newlineSo, (2x+1)/(x+2)=2−3/(x+2)(2x+1)/(x+2) = 2 - 3/(x+2).
  2. Split Integral: Now we can split the integral into two parts. ∫2x+1x+2 dx=∫2 dx−∫3x+2 dx\int\frac{2x+1}{x+2}\,dx = \int 2 \,dx - \int \frac{3}{x+2} \,dx.
  3. Integrate Separately: Integrate each part separately.\newlineThe integral of 2 dx2 \, dx is 2x2x.\newlineThe integral of 3(x+2) dx\frac{3}{(x+2)} \, dx is 3ln⁡∣x+2∣3\ln|x+2|.
  4. Combine Integrals: Combine the two integrals.\newlineSo, ∫2x+1x+2 dx=2x−3ln⁡∣x+2∣+C\int \frac{2x+1}{x+2}\,dx = 2x - 3\ln|x+2| + C.
  5. Check Answer Choices: Check the answer choices.\newlineThe correct answer matches with (A) 2x−3ln⁡∣x+2∣+C2x - 3\ln|x+2| + C.

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