AP Calculus AB 6.10 Integrating Functions Using Long Division and Completing the Square - MCQs - Exam Style Questions
No-Calc Question
\(\displaystyle \int \dfrac{dx}{x^{2}-6x+10}=\ \ ?\)
(A) \(\ln|x^{2}-6x+10|+C\)
(B) \(\dfrac{1}{2x-6}\ln|x^{2}-6x+10|+C\)
(C) \(\dfrac{1}{2}\arctan(x-3)+C\)
(D) \(\arctan(x-3)+C\)
▶️ Answer/Explanation
Complete the square: \(x^{2}-6x+10=(x-3)^{2}+1\).
Use \(\displaystyle\int \dfrac{dx}{u^{2}+1}=\arctan u + C\) with \(u=x-3\).
Therefore \(\displaystyle\int\dfrac{dx}{x^{2}-6x+10}=\arctan(x-3)+C\).
✅ Answer: (D)