AP Calculus BC 6.10 Integrating Functions Using Long Division and Completing the Square - MCQs - Exam Style Questions
No-CalcQuestion
Evaluate \( \displaystyle \int \frac{x^{2}+4x-1}{x+2}\,dx \).
(A) \( \tfrac{1}{3}x^{3}+2x^{2}-x-\dfrac{x}{x^{2}+2x}+C \)
(B) \( x+2-5\ln|x+2|+C \)
(C) \( \tfrac{1}{2}x^{2}+2x-5\ln|x+2|+C \)
(D) \( \tfrac{1}{2}x^{2}+2x+\dfrac{5}{x+2}+C \)
▶️ Answer/Explanation
Detailed solution
Long divide: \[ \frac{x^{2}+4x-1}{x+2}=(x+2)-\frac{5}{x+2}. \] Integrate: \[ \int(x+2)\,dx=\tfrac{1}{2}x^{2}+2x,\qquad \int\!\left(-\frac{5}{x+2}\right)dx=-5\ln|x+2|. \] \[ \Rightarrow \int \frac{x^{2}+4x-1}{x+2}\,dx=\tfrac{1}{2}x^{2}+2x-5\ln|x+2|+C. \] ✅ Answer: (C)
No-Calc Question
\(\displaystyle \int \frac{dx}{x^{2}+4x+5}=\) ?
(A) \(\tan^{-1}(x+2)+C\)
(B) \(\ln|x^{2}+4x+5|+C\)
(C) \(\dfrac{1}{2x+4}\,\ln|x^{2}+4x+5|+C\)
(D) \(\dfrac{1}{\tfrac{1}{3}x^{3}+2x^{2}+5x}+C\)
(B) \(\ln|x^{2}+4x+5|+C\)
(C) \(\dfrac{1}{2x+4}\,\ln|x^{2}+4x+5|+C\)
(D) \(\dfrac{1}{\tfrac{1}{3}x^{3}+2x^{2}+5x}+C\)
▶️ Answer/Explanation
Detailed solution
Complete the square: \(x^{2}+4x+5=(x+2)^{2}+1\).
Then \(\displaystyle \int \frac{dx}{(x+2)^{2}+1}=\tan^{-1}(x+2)+C\).
✅ Answer: (A)