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

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

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

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

Flashcard 3: What is the integral of $\csc(x) \cot(x)$?

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

Flashcard 4: 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 5: 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 6: What is the integral of $\sec(x) \tan(x)$?

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

Flashcard 7: What is the integral of $\csc^2(x)$?

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

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

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

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

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

Flashcard 10: What is the integral of $\csc(x) \cot(x)$?

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

Flashcard 11: Evaluate: $\int_0^3 (x^2) \, dx$.

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

Flashcard 12: State the integral of $\sec^2(x)$.

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

Flashcard 13: Calculate $\int_1^2 (2x^2 - 3) \, dx$.

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

Flashcard 14: What is the integral of $\sin(x)$?

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

Flashcard 15: State the integration by parts formula.

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

Flashcard 16: What is the integral of $\csc^2(x)$?

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

Flashcard 17: State the integral of $\sec^2(x)$.

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

Flashcard 18: What is the integral of $e^x$?

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

Flashcard 19: What is the integral of $e^x$?

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

Flashcard 20: Integrate: $\int (2x + 3) \, dx$.

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

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

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

Flashcard 22: What is the integral of $\frac{1}{1+x^2}$?

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

Flashcard 23: Evaluate: $\int_0^2 (4x^3) \, dx$.

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

Flashcard 24: What is the integral of $\frac{1}{\cos^2(x)}$?

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

Flashcard 25: State the integral of $\frac{1}{x^2 + 1}$.

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

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

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

Flashcard 27: What is $C$ in an antiderivative?

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

Flashcard 28: 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 29: State the integral of $\frac{1}{x^2 + 1}$.

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

Flashcard 30: Evaluate: $\int_0^3 (x^2) \, dx$.

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

Flashcard 31: What is the integral of $\frac{1}{\sqrt{1-x^2}}$?

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

Flashcard 32: What is the antiderivative of $\frac{1}{x}$?

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

Flashcard 33: What is the antiderivative of $f(x) = e^x$?

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

Flashcard 34: 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 35: What is $C$ in an antiderivative?

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

Flashcard 36: 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 37: 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 38: Evaluate: $\int_0^2 (4x^3) \, dx$.

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

Flashcard 39: 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 40: What is the antiderivative of $\frac{1}{x}$?

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

Flashcard 41: What is the integral of $\cos(x)$?

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

Flashcard 42: 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 43: What is the integral of $\frac{1}{x}$?

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

Flashcard 44: What is the antiderivative of $f(x) = e^x$?

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

Flashcard 45: 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 46: What is the integral of $\tan(x)$?

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

Flashcard 47: State the integration by parts formula.

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

Flashcard 48: What is the integral of $\frac{1}{x}$?

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

Flashcard 49: What is the integral of $\frac{1}{1+x^2}$?

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

Flashcard 50: Integrate: $\int (2x + 3) \, dx$.

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

Flashcard 51: What is the integral of $\frac{1}{\sqrt{1-x^2}}$?

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

Flashcard 52: What is the substitution method in integration?

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