Home / AP Calculus AB 6.8 Finding Antiderivatives and Indefinite Integrals – MCQs

AP Calculus AB 6.8 Finding Antiderivatives and Indefinite Integrals - MCQs - Exam Style Questions

AP Calculus AB-MCQs and Answers - All Topics

No-Calc Question

\(\ \displaystyle \int \frac{x^{2}}{4}\,dx=\)
(A) \( \dfrac{x}{2}+C\)
(B) \( \dfrac{x^{3}}{12}+C\)
(C) \( \dfrac{x^{3}}{4}+C\)
(D) \( \dfrac{3x^{3}}{4}+C\)
▶️ Answer/Explanation

Pull out \(\frac{1}{4}\).
\(\displaystyle \int \frac{x^{2}}{4}\,dx=\frac{1}{4}\int x^{2}\,dx\).
\(\displaystyle =\frac{1}{4}\cdot \frac{x^{3}}{3}+C=\frac{x^{3}}{12}+C\).

✅ Answer: (B)

No-Calc Question

If \(f'(x)=\dfrac{2}{x}\) and \(f(\sqrt{e})=5\), then \(f(e)=\)

(A) \(2\)
(B) \(\ln 25\)
(C) \(5+\dfrac{2}{e}-\dfrac{2}{e^{2}}\)
(D) \(6\)
(E) \(25\)

▶️ Answer/Explanation

Integrate \(f'(x)=\dfrac{2}{x}\): \(f(x)=2\ln|x|+C\).
Use \(f(\sqrt{e})=5\): \(2\ln(\sqrt{e})+C=5\).
\(\ln(\sqrt{e})=\tfrac{1}{2}\ln e=\tfrac{1}{2}\). Hence \(2\times\tfrac{1}{2}+C=5\Rightarrow C=4\).
Now \(f(e)=2\ln e+4=2+4=6\).
✅ Answer: (D)

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