AP Calculus BC Flashcards: Exploring Accumulations Of Change

What this deck covers

This deck focuses on Exploring Accumulations Of Change, giving you a quick way to review the definitions, rules, and examples that matter most for AP Calculus BC.

How to use these flashcards

Work through these flashcards in short sessions. Try to answer each prompt before flipping the card, then revisit any cards you miss until the explanation feels automatic.

Flashcard 1: What is the integral of $\frac{1}{\sqrt{1-x^2}}$?

Answer: $\arcsin(x) + C$. Standard inverse trigonometric integration formula.

Flashcard 2: What is the integral of $\csc^2(x)$?

Answer: $-\cot(x) + C$. Standard trigonometric integration formula.

Flashcard 3: What is the integral of $\frac{1}{\cos^2(x)}$?

Answer: $\tan(x) + C$. Since $\frac{1}{\cos^2(x)} = \sec^2(x)$.

Flashcard 4: What is the antiderivative of $\frac{1}{x}$?

Answer: $\ln|x| + C$. Natural logarithm is the antiderivative of $\frac{1}{x}$.

Flashcard 5: Integrate: $\int (2x + 3) \, dx$.

Answer: $x^2 + 3x + C$. Apply power rule to each term separately.

Flashcard 6: State the integration by parts formula.

Answer: $\int u \, dv = uv - \int v \, du$. Method for integrating products of functions.

Flashcard 7: Evaluate: $\int_0^2 (4x^3) \, dx$.

Answer: $16$. Evaluate $x^4$ from 0 to 2 gives $16 - 0 = 16$.

Flashcard 8: What is the integral of $\sec(x) \tan(x)$?

Answer: $\sec(x) + C$. Standard trigonometric integration formula.

Flashcard 9: What is the integral of $\frac{1}{1+x^2}$?

Answer: $\arctan(x) + C$. Standard inverse trigonometric integration formula.

Flashcard 10: Evaluate: $\int_0^3 (x^2) \, dx$.

Answer: $9$. Evaluate $\frac{x^3}{3}$ from 0 to 3 gives $\frac{27}{3} = 9$.

Flashcard 11: Integrate: $\int 5x^4 \, dx$.

Answer: $x^5 + C$. Apply power rule: $\int 5x^4 dx = 5 \cdot \frac{x^5}{5} = x^5$.

Flashcard 12: What is the integral of $\tan(x)$?

Answer: $-\ln|\cos(x)| + C$. Using substitution $u = \cos(x)$.

Flashcard 13: What is $C$ in an antiderivative?

Answer: A constant of integration. Represents the family of antiderivatives for indefinite integrals.

Flashcard 14: What does the definite integral of a positive function represent?

Answer: The area under the curve $f(x)$ from $x = a$ to $x = b$. Geometric interpretation of definite integrals for positive functions.

Flashcard 15: What is the integral of $e^x$?

Answer: $e^x + C$. The exponential function is its own antiderivative.

Flashcard 16: State the integral of $\sec^2(x)$.

Answer: $\tan(x) + C$. Standard trigonometric integration formula.

Flashcard 17: Calculate $\int_1^2 (2x^2 - 3) \, dx$.

Answer: $\frac{5}{3}$. Evaluate $\frac{2x^3}{3} - 3x$ from 1 to 2.

Flashcard 18: Calculate: $\int_0^1 (3x + 2) \, dx$.

Answer: $3.5$. Evaluate $\frac{3x^2}{2} + 2x$ from 0 to 1 gives $\frac{3}{2} + 2 = 3.5$.

Flashcard 19: What is the antiderivative of $f(x) = e^x$?

Answer: $F(x) = e^x + C$. The exponential function is its own antiderivative.

Flashcard 20: What is the substitution method in integration?

Answer: Substitute $u = g(x)$ to simplify the integral. Technique for simplifying complex integrals by changing variables.

Flashcard 21: What is the integral of $\cos(x)$?

Answer: $\sin(x) + C$. Standard trigonometric integration formula.

Flashcard 22: What is the integral of $\frac{1}{x}$?

Answer: $\ln|x| + C$. Natural logarithm is the antiderivative of $\frac{1}{x}$.

Flashcard 23: State the integral of $\frac{1}{x^2 + 1}$.

Answer: $\arctan(x) + C$. Standard inverse trigonometric integration formula.

Flashcard 24: State the power rule for integration.

Answer: $\int x^n \, dx = \frac{x^{n+1}}{n+1} + C$, $n \neq -1$. Fundamental integration rule for polynomial functions.

Flashcard 25: What is the integral of $\sin(x)$?

Answer: $-\cos(x) + C$. Standard trigonometric integration formula.

Flashcard 26: What is the integral of $\csc(x) \cot(x)$?

Answer: $-\csc(x) + C$. Standard trigonometric integration formula.

Flashcard 27: What is the integral of $\csc^2(x)$?

Answer: $-\cot(x) + C$. Standard trigonometric integration formula.

Flashcard 28: What is $C$ in an antiderivative?

Answer: A constant of integration. Represents the family of antiderivatives for indefinite integrals.

Flashcard 29: What does the definite integral of a positive function represent?

Answer: The area under the curve $f(x)$ from $x = a$ to $x = b$. Geometric interpretation of definite integrals for positive functions.

Flashcard 30: What is the integral of $e^x$?

Answer: $e^x + C$. The exponential function is its own antiderivative.