AP Calculus BC Practice Quiz: Finding Antiderivatives and Indefinite Integrals: Basic Rules and Notation

Written by AP Content Team, Verified for 2026 AP Exams, Last updated: May 2026

Test your understanding with short quizzes. This quiz has 10 questions to check your progress.

Question 1 of 10

If $\int f(x) dx = F(x) + C$, which of the following statements correctly describes the relationship between $f(x)$ and $F(x)$?

A)$F'(x) = f(x)$B)$f'(x) = F(x)$C)$F(x) = f(x)$D)$F(x) = f(x) + C$

All Questions (10)

If $\int f(x) dx = F(x) + C$, which of the following statements correctly describes the relationship between $f(x)$ and $F(x)$?

A) $F'(x) = f(x)$

B) $f'(x) = F(x)$

C) $F(x) = f(x)$

D) $F(x) = f(x) + C$

Correct Answer: A

By the definition of an indefinite integral, the integral of a function $f(x)$ is a new function $F(x)$ (called the antiderivative) whose derivative is the original function $f(x)$. [cite: 2671]

In the expression for an indefinite integral, $\int f(x) dx = F(x) + C$, what is the significance of the constant $C$?

A) It represents a specific, required value for the integral to be correct.

B) It indicates that the antiderivative is a family of functions whose graphs are vertical translations of each other.

C) It is a variable that depends on the value of x.

D) It signifies that the integral cannot be determined exactly.

Correct Answer: B

The constant of integration, $C$, represents the fact that there are infinitely many functions that have the same derivative. These functions are all vertical shifts of one another, forming a family of functions. For any constant $C$, the derivative of $F(x) + C$ is $F'(x) + 0 = f(x)$. [cite: 2671]

Which of the following is an antiderivative of the function $f(x) = \cos(x)$?

A) $-\sin(x)$

B) $-\cos(x)$

C) $\sin(x)$

D) $\sec^2(x)$

Correct Answer: C

To find an antiderivative, we must find a function whose derivative is $\cos(x)$. Using knowledge of differentiation rules, we know that the derivative of $\sin(x)$ is $\cos(x)$. Therefore, $\sin(x)$ is an antiderivative of $\cos(x)$. [cite: 2670]

The process of finding an indefinite integral is fundamentally based on which mathematical concept?

A) Applying the rules of limits.

B) Solving algebraic equations.

C) Reversing the rules of differentiation.

D) Calculating the area under a curve.

Correct Answer: C

Finding an antiderivative is the inverse process of differentiation. Therefore, the rules for finding antiderivatives are derived directly from the rules for finding derivatives. [cite: 2671]

If $F'(x) = G'(x)$ for all $x$ in an interval, which of the following must be true for the functions $F(x)$ and $G(x)$ on that interval?

A) $F(x) = G(x)$

B) $F(x)$ and $G(x)$ are parallel lines.

C) $F(x) - G(x) = C$ for some constant $C$.

D) $\int F(x) dx = \int G(x) dx$

Correct Answer: C

If two functions have the same derivative, they must differ by a constant. This is the principle behind the constant of integration, $C$, in an indefinite integral. For example, the derivatives of $x^2$ and $x^2+5$ are both $2x$. [cite: 2671]

What is the indefinite integral $\int 5x^4 dx$?

A) $20x^3 + C$

B) $x^5 + C$

C) $5x^5 + C$

D) $x^4 + C$

Correct Answer: B

Using knowledge of the power rule for derivatives in reverse, we look for a function whose derivative is $5x^4$. The derivative of $x^5$ is $5x^4$. Therefore, the indefinite integral is $x^5 + C$. [cite: 2670, 2671]

Consider the function $f(x) = \sin(x^2)$. Which of the following statements about its antiderivative, $F(x) = \int \sin(x^2) dx$, is correct?

A) $F(x) = -\cos(x^2) + C$

B) $F(x) = 2x\cos(x^2) + C$

C) The antiderivative $F(x)$ does not exist.

D) The antiderivative $F(x)$ exists but does not have a closed-form expression using elementary functions.

Correct Answer: D

While an antiderivative for $\sin(x^2)$ exists (since it's a continuous function), it cannot be expressed using a finite combination of elementary functions (polynomials, trigonometric, exponential, logarithmic functions). This is an example of a function that does not have a closed-form antiderivative. [cite: 2672]

The expression $\int f(x) dx$ is best described as:

A) The derivative of $f(x)$.

B) The indefinite integral of $f(x)$.

C) A definite integral of $f(x)$.

D) The value of $f(x)$ at a specific point.

Correct Answer: B

The notation $\int f(x) dx$ is the standard notation for the indefinite integral of the function $f$ with respect to $x$. It represents the family of all antiderivatives of $f$. [cite: 2671]

Given that $F(x) = e^x + x^2$ is an antiderivative of a function $f(x)$, which of the following is also an antiderivative of $f(x)$?

A) $e^x + x^2 - 5$

B) $5(e^x + x^2)$

C) $e^x + 2x$

D) $e^{x-5} + (x-5)^2$

Correct Answer: A

If $F(x)$ is an antiderivative of $f(x)$, then any function of the form $F(x) + C$, where $C$ is any constant, is also an antiderivative. This is because the derivative of the constant $C$ is zero. In this case, $e^x + x^2 - 5$ differs from the original antiderivative by a constant, -5. [cite: 2671]

How would one verify that $F(x) = \frac{1}{3}x^3 + \ln|x|$ is a correct antiderivative for $f(x) = x^2 + \frac{1}{x}$?

A) By integrating $F(x)$.

B) By evaluating both functions at $x=1$.

C) By differentiating $F(x)$ to see if the result is $f(x)$.

D) By confirming that $F(x)$ does not have a closed-form.

Correct Answer: C

The definition of an antiderivative states that $F(x)$ is an antiderivative of $f(x)$ if and only if $F'(x) = f(x)$. Therefore, the way to check if an antiderivative is correct is to perform differentiation on the proposed antiderivative. [cite: 2670, 2671]