AP Calculus AB: Antiderivatives and Basic Integration Rules (Drill 1)

Question 1. What is \( \int (4x^3 - 6x + 5)\,dx \)?

• A) \( 12x^2 - 6 + C \)
• B) \( 4x^4 - 6x^2 + 5x + C \)
• C) \( x^4 - 6x^2 + 5x + C \)
• D) \( x^4 - 3x^2 + 5x + C \) ✓

Explanation: Choice D is correct. Applying the power rule \( \int x^n\,dx = \dfrac{x^{n+1}}{n+1} + C \) term by term: \( \int 4x^3\,dx = x^4 \), \( \int -6x\,dx = -3x^2 \), and \( \int 5\,dx = 5x \), giving \( x^4 - 3x^2 + 5x + C \). Choice A is incorrect because the student differentiated the integrand rather than integrating it, producing \( 12x^2 - 6 \). Choice B is incorrect because the student raised the exponent on \( 4x^3 \) to 4 but did not divide by the new exponent, writing \( 4x^4 \) instead of \( x^4 \). Choice C is incorrect because the student applied the power rule to \( -6x \) but forgot to divide by the new exponent of 2, writing \( -6x^2 \) instead of \( -3x^2 \).

Question 2. What is \( \int \left( e^x + \dfrac{3}{x} \right) dx \)?

• A) \( e^x + \dfrac{3}{x^2} + C \)
• B) \( xe^x + 3\ln|x| + C \)
• C) \( e^x + 3\ln|x| + C \) ✓
• D) \( e^x - \dfrac{3}{x^2} + C \)

Explanation: Choice C is correct. The antiderivative of \( e^x \) is \( e^x \), and the antiderivative of \( \dfrac{3}{x} \) is \( 3\ln|x| \), giving \( e^x + 3\ln|x| + C \). Choice A is incorrect because the student applied the power rule to \( x^{-1} \), which fails since \( \dfrac{x^{n+1}}{n+1} \) is undefined when \( n = -1 \); the antiderivative of \( x^{-1} \) is \( \ln|x| \), a fact that must be memorized separately. Choice B is incorrect because the student treated \( \int e^x\,dx \) as an integration by parts problem, incorrectly obtaining \( xe^x \); since \( \dfrac{d}{dx}[e^x] = e^x \), its antiderivative is simply \( e^x \). Choice D is incorrect because the student differentiated \( \dfrac{3}{x} \) rather than integrating it, producing \( -\dfrac{3}{x^2} \) (the derivative of \( 3x^{-1} \)).

Question 3. Which of the following is an antiderivative of \( f(x) = \sec^2 x - \sin x \)?

• A) \( \tan x + \cos x \) ✓
• B) \( \tan x - \cos x \)
• C) \( 2\sec x\tan x + \cos x \)
• D) \( \sec x + \cos x \)

Explanation: Choice A is correct. The standard antiderivative formulas give \( \int \sec^2 x\,dx = \tan x \) and \( \int -\sin x\,dx = \cos x \), so \( F(x) = \tan x + \cos x \). Verification: \( \dfrac{d}{dx}[\tan x + \cos x] = \sec^2 x - \sin x \). Choice B is incorrect because of a sign error on the second term: \( \int -\sin x\,dx = \cos x \), not \( -\cos x \), because \( \dfrac{d}{dx}[\cos x] = -\sin x \). This is the most common error on this type of question. Choice C is incorrect because \( 2\sec x\tan x \) is the derivative of \( \sec^2 x \), not its antiderivative, the student differentiated the first term rather than integrating it. Choice D is incorrect because \( \sec x \) is not the antiderivative of \( \sec^2 x \); the student may have confused the relationship between \( \sec x \) and its derivative \( \sec x\tan x \), or simply reduced the exponent rather than applying the correct formula.

Question 4. If \( F'(x) = 3x^2 - 4x \) and \( F(1) = 5 \), what is \( F(x) \)?

• A) \( x^3 - 2x^2 - 1 \)
• B) \( x^3 - 2x^2 + 5 \)
• C) \( 6x - 4 \)
• D) \( x^3 - 2x^2 + 6 \) ✓

Explanation: Choice D is correct. Integrating: \( F(x) = x^3 - 2x^2 + C \). Applying \( F(1) = 5 \): \( 1 - 2 + C = 5 \Rightarrow -1 + C = 5 \Rightarrow C = 6 \), so \( F(x) = x^3 - 2x^2 + 6 \). Choice A is incorrect because of a sign error when solving for \( C \): from \( -1 + C = 5 \) the correct result is \( C = 6 \), not \( C = -1 \). Choice B is incorrect because the student set \( C = F(1) = 5 \) directly without substituting \( x = 1 \) into the antiderivative first, a very common initial value problem error that skips the solving step entirely. Choice C is incorrect because the student differentiated \( F'(x) \) again rather than integrating it, producing \( F''(x) = 6x - 4 \).

Question 5. A student claims that \( \int (2x + 1)\,dx = x^2 + x \) is a complete and valid answer. Which of the following best describes the error?

• A) The power rule was applied incorrectly; \( \int 2x\,dx \) should equal \( x^2 + 2 \), not \( x^2 \).
• B) The answer represents only one function in the family of antiderivatives; omitting \( +C \) is incorrect because every function of the form \( x^2 + x + C \) has derivative \( 2x + 1 \). ✓
• C) The antiderivative of a sum cannot be computed term by term; \( 2x + 1 \) must be integrated as a single expression using substitution.
• D) There is no error; \( x^2 + x \) is a complete answer to an indefinite integral.

Explanation: Choice B is correct. The computation \( x^2 + x \) is one valid antiderivative, but an indefinite integral represents an entire family of functions \( x^2 + x + C \), where \( C \) is any real constant. Every function of this form has derivative \( 2x + 1 \), so omitting \( +C \) is genuinely incomplete. Choice A is incorrect because \( \int 2x\,dx = x^2 \) is right (via power rule: \( \frac{2x^2}{2} = x^2 \)); no computation error was made. Choice C is incorrect because the antiderivative of a sum is the sum of the antiderivatives; this is a valid linearity property, not a restriction requiring substitution. Choice D is incorrect because omitting \( +C \) is a real error on an indefinite integral; a specific antiderivative is only acceptable when an initial condition is provided to determine \( C \).