AP Calculus BC — Practice Questions flashcards

6 of 50 questions free to study here · 5 topics · the whole deck is built into Willys

Multiple-choice practice questions, one best answer each, on integration by parts, improper integrals, Euler's method, logistic growth, parametric, polar and vector calculus, and series.

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Practice deck · part of AP Calculus BC · written against: College Board AP Calculus BC Course and Exam Description (BC-only topics and Unit 10 series). Updated 2026-10-10.

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Which of the following is $\int x e^{2x} dx$ ⁠?

A) $\frac{x}{2}e^{2x} - \frac{1}{4}e^{2x} + C$
B) $\frac{x}{2}e^{2x} + \frac{1}{4}e^{2x} + C$
C) $\frac{x}{2}e^{2x} - \frac{1}{2}e^{2x} + C$
D) $x e^{2x} - \frac{1}{2}e^{2x} + C$

A) $\frac{x}{2}e^{2x} - \frac{1}{4}e^{2x} + C$

With $u = x$ and $dv = e^{2x} dx$ ⁠, $v = \frac{1}{2}e^{2x}$ ⁠, so the result is $\frac{x}{2}e^{2x} - \int \frac{1}{2}e^{2x} dx$ ⁠. The second integral brings another factor of $\frac{1}{2}$ ⁠, giving $\frac{1}{4}$ ⁠, not $\frac{1}{2}$ ⁠.
Integration and Accumulation of Change › Integration by Parts and Partial Fractions
What is the value of $\int_1^e x \ln x dx$ ⁠?

A) $\frac{e^2}{4}$
B) $\frac{e^2 - 1}{4}$
C) $\frac{e^2 + 1}{4}$
D) $\frac{e^2 + 1}{2}$

C) $\frac{e^2 + 1}{4}$

Take $u = \ln x$ ⁠, $dv = x dx$ ⁠: the antiderivative is $\frac{x^2}{2}\ln x - \frac{x^2}{4}$ ⁠. It is $\frac{e^2}{4}$ at $e$ and $-\frac{1}{4}$ at 1, so the value is $\frac{e^2 + 1}{4}$ ⁠; dropping the lower limit gives just $\frac{e^2}{4}$ ⁠.
Integration and Accumulation of Change › Integration by Parts and Partial Fractions
Let $y = f(x)$ solve $\frac{dy}{dx} = 2x - y$ with $f(1) = 3$ ⁠. Using Euler's method with two steps of equal size starting at $x = 1$ ⁠, what is the approximation of $f(2)$ ⁠?

A) $2.75$
B) $3.25$
C) $2$
D) $2.5$

A) $2.75$

With $h = 0.5$ ⁠: the slope at $(1, 3)$ is $-1$ ⁠, giving $2.5$ at $x = 1.5$ ⁠; the slope there is $3 - 2.5 = 0.5$ ⁠, giving $2.5 + 0.25 = 2.75$ ⁠. One step of size 1 would give 2, and stopping after the first step gives 2.5.
Differential Equations › Euler's Method and Logistic Models
Which integral gives the length of the curve $y = \sin(2x)$ from $x = 0$ to $x = \frac{\pi}{2}$ ⁠?

A) $\int_0^{\pi/2} \sqrt{1 + \sin^2(2x)} dx$
B) $\int_0^{\pi/2} \sqrt{1 + 4\cos^2(2x)} dx$
C) $\int_0^{\pi/2} \left(1 + 4\cos^2(2x)\right) dx$
D) $\int_0^{\pi/2} \sqrt{1 + \cos^2(2x)} dx$

B) $\int_0^{\pi/2} \sqrt{1 + 4\cos^2(2x)} dx$

Arc length is $\int \sqrt{1 + \left(y'\right)^2} dx$ and $y' = 2\cos(2x)$ ⁠, so $\left(y'\right)^2 = 4\cos^2(2x)$ ⁠. Forgetting the chain-rule factor 2 drops the 4, and leaving out the square root gives a different quantity.
Applications of Integration › Arc Length
A curve is given by $x = t^2 + 1$ and $y = t^3 - 2t$ ⁠. What is the slope of the tangent line at $t = 2$ ⁠?

A) $10$
B) $\frac{2}{5}$
C) $\frac{4}{5}$
D) $\frac{5}{2}$

D) $\frac{5}{2}$

$\frac{dy}{dx} = \frac{3t^2 - 2}{2t}$ ⁠, which is $\frac{10}{4} = \frac{5}{2}$ at $t = 2$ ⁠. Using $\frac{dy}{dt}$ alone gives 10, and inverting the quotient gives $\frac{2}{5}$ ⁠.
Parametric Polar and Vector-Valued Functions › Parametric Equations
What is the sum of $\sum_{n=0}^{\infty} \frac{2^{n+1}}{3^n}$ ⁠?

A) $2$
B) $4$
C) $6$
D) $3$

C) $6$

The first term ( $n = 0$ ⁠) is 2 and the ratio is $\frac{2}{3}$ ⁠, so the sum is $\frac{2}{1 - 2/3} = 6$ ⁠. Using 1 as the first term gives 3, and starting the count at the second term gives 4.
Infinite Sequences and Series › Sequences and Geometric Series

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About these cards

Review status: Second-reader reviewed, with corrections applied.

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Topic outline follows the BC-only topics of the College Board AP Calculus AB and BC Course and Exam Description (effective Fall 2020), Units 6–10 (the AP name is used nominatively; no course or exam text reproduced).
OpenStax Calculus Volumes 1 and 2, consulted as references; no text copied.
Every answer was verified with a computer algebra script.

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