AP Calculus BC Practice Quiz: Selecting Techniques for Antidifferentiation
Written by AP Content Team, Verified for 2026 AP Exams, Last updated: May 2026
Test your understanding with short quizzes. This quiz has 7 questions to check your progress.
Question 1 of 7
All Questions (7)
A) Integration by Parts
B) u-Substitution
C) Partial Fraction Decomposition
D) Trigonometric Substitution
Correct Answer: B
The integrand contains a composite function, sin(x²), and the derivative of the inner function, x², is 2x. A multiple of this derivative (the 'x' term) is present as a factor in the integrand. This structure is a clear indicator for using u-substitution with u = x².
A) u-Substitution
B) Recognizing it as a basic antiderivative rule
C) Integration by Parts
D) Partial Fraction Decomposition
Correct Answer: C
The integrand is a product of two different types of functions: a polynomial function (x) and an exponential function (e^(3x)). This structure is a classic indicator for Integration by Parts, where one function is chosen as 'u' and the other as 'dv'.
A) Partial Fraction Decomposition
B) Integration by Parts
C) u-Substitution with u = x² - 2x - 3
D) Recognizing it as the derivative of an inverse trigonometric function
Correct Answer: A
The integrand is a rational function where the denominator, x² - 2x - 3, can be factored into (x - 3)(x + 1). This indicates that the fraction can be decomposed into simpler fractions, making Partial Fraction Decomposition the correct approach.
A) u-Substitution with u = x² + 16
B) Partial Fraction Decomposition
C) Integration by Parts
D) Recognizing it as a form leading to an inverse tangent function
Correct Answer: D
The integral is in the form ∫ 1 / (x² + a²) dx, where a = 4. This is the standard form for the antiderivative involving the inverse tangent (arctan) function. While trigonometric substitution could also be used, recognizing the inverse tangent form is more direct.
A) u-Substitution
B) Integration by Parts
C) Direct application of the power rule
D) Partial Fraction Decomposition
Correct Answer: B
Although ln(x) is a single function, Integration by Parts is the correct method. We can set u = ln(x) and dv = dx. This allows us to differentiate ln(x) to get 1/x and integrate dv to get x, resulting in a simpler integral to solve.
A) Use u-substitution with u = x² + 9 on the entire integrand.
B) Use integration by parts with u = 2x + 3 and dv = 1/(x² + 9) dx.
C) Split the integrand into two separate fractions and apply a different technique to each part.
D) Use partial fraction decomposition after factoring the denominator.
Correct Answer: C
The most effective method is to split the integral into two: ∫ (2x / (x² + 9)) dx + ∫ (3 / (x² + 9)) dx. The first part can be solved using u-substitution (u = x² + 9), and the second part is in the form of an inverse tangent function. No single method works for the original combined integrand.
A) Integration by Parts
B) Partial Fraction Decomposition
C) u-Substitution after using a Pythagorean identity
D) Trigonometric Substitution
Correct Answer: C
For integrals with powers of sine and cosine, the strategy is to use a Pythagorean identity to set up a u-substitution. Here, since the power of sine is odd, we can rewrite sin³(x) as sin²(x)sin(x), then use the identity sin²(x) = 1 - cos²(x). This transforms the integral into ∫ (1 - cos²(x))cos²(x)sin(x) dx, which is ready for a u-substitution with u = cos(x).