AP Calculus AB Practice Quiz: Selecting Techniques for Antidifferentiation
Written by AP Content Team, Verified for 2026 AP Exams, Last updated: July 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, cos(x^2), and the derivative of the inner function (x^2) is 2x, a constant multiple of which (x) is also present as a factor in the integrand. This structure, f(g(x))g'(x), is the classic indicator for using u-substitution with u = x^2.
A) u-substitution
B) Integration by Parts
C) Partial Fraction Decomposition
D) Trigonometric Substitution
Correct Answer: B
The integrand is a product of two fundamentally different types of functions: an algebraic function (x) and an exponential function (e^(3x)). This is a standard scenario for Integration by Parts, where one function can be simplified by differentiation (x) and the other can be easily integrated (e^(3x)).
A) u-substitution with u = x^2 - 2x - 3
B) Integration by Parts
C) Partial Fraction Decomposition
D) Trigonometric Substitution
Correct Answer: C
The integrand is a rational function where the denominator, x^2 - 2x - 3, can be factored into (x - 3)(x + 1). This structure allows the fraction to be decomposed into a sum of simpler fractions, which is the method of Partial Fraction Decomposition.
A) u-substitution
B) Integration by Parts
C) Partial Fraction Decomposition
D) Trigonometric Substitution
Correct Answer: D
The presence of the term sqrt(9 - x^2), which is of the form sqrt(a^2 - x^2), is a strong indicator for using Trigonometric Substitution. Specifically, the substitution x = 3sin(θ) would be used to simplify the radical.
A) Partial Fraction Decomposition, because the integrand is a rational function.
B) Integration by Parts, with u = 1 and dv = 1 / (x^2 + 4x + 13) dx.
C) Completing the square on the denominator to transform the integrand into a form suitable for an arctangent rule.
D) u-substitution, with u = x^2 + 4x + 13.
Correct Answer: C
The denominator x^2 + 4x + 13 is an irreducible quadratic. Therefore, Partial Fraction Decomposition is not possible. A u-substitution with u as the denominator fails because du = (2x + 4)dx, which is not present in the numerator. The most effective strategy is to complete the square in the denominator: x^2 + 4x + 13 = (x^2 + 4x + 4) + 9 = (x + 2)^2 + 3^2. This transforms the integral into a standard form that yields an arctangent function after a simple u-substitution.
A) u-substitution
B) Integration by Parts
C) Trigonometric Substitution
D) Partial Fraction Decomposition
Correct Answer: B
While the integral of arctan(x) is not immediately obvious, it can be found using Integration by Parts. This is a classic case where the integrand is treated as a product of itself and 1. By setting u = arctan(x) and dv = dx, we can differentiate arctan(x) to get a simpler algebraic expression and integrate dv easily.
A) First use Integration by Parts, then use u-substitution.
B) First use u-substitution, then use Integration by Parts.
C) Use only Partial Fraction Decomposition.
D) Use only Trigonometric Substitution.
Correct Answer: B
The integrand e^(sqrt(x)) is difficult to work with directly. The first step is to simplify the argument of the exponential function with a u-substitution. Let u = sqrt(x), then x = u^2 and dx = 2u du. The integral transforms into ∫ 2u * e^u du. This new integral is a product of an algebraic function (2u) and an exponential function (e^u), which is a standard form for Integration by Parts.