AP Calculus AB Flashcards: Applying Properties Of Definite Integrals

What this deck covers

This deck focuses on Applying Properties Of Definite Integrals, giving you a quick way to review the definitions, rules, and examples that matter most for AP Calculus AB.

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 $\text{cos}(x)$?

Answer: $\text{sin}(x) + C$. Standard antiderivative: derivative of sine is cosine.

Flashcard 2: Evaluate $\frac{d}{dx}\bigg(\frac{1}{\text{csc}(x)}\bigg)$.

Answer: $\text{cot}(x)\text{csc}(x)$. Chain rule on $\sin(x) = (\csc(x))^{-1}$ gives $\cos(x)$.

Flashcard 3: What is the antiderivative of $\text{sin}(x)$?

Answer: $-\text{cos}(x) + C$. Standard antiderivative: derivative of negative cosine is sine.

Flashcard 4: What is the antiderivative of $\text{sin}(x)$?

Answer: $-\text{cos}(x) + C$. Standard antiderivative: derivative of negative cosine is sine.

Flashcard 5: What is the antiderivative of $e^x$?

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

Flashcard 6: Evaluate $\frac{d}{dx}\bigg(\frac{1}{\text{cos}(x)}\bigg)$.

Answer: $\frac{\text{sin}(x)}{\text{cos}^2(x)}$. Chain rule on $\sec(x) = (\cos(x))^{-1}$ gives $\sec(x)\tan(x)$.

Flashcard 7: What is the value of $\frac{d}{dx}\bigg(\frac{1}{x}\bigg)$?

Answer: $-\frac{1}{x^2}$. Power rule applied to $x^{-1}$.

Flashcard 8: What is the integral of $x^n$ with respect to $x$?

Answer: $\frac{x^{n+1}}{n+1} + C$ for $n \neq -1$. Power rule for integration increases exponent by 1.

Flashcard 9: What is the integral of $\text{cos}(x)$?

Answer: $\text{sin}(x) + C$. Standard antiderivative: derivative of sine is cosine.

Flashcard 10: Evaluate $\frac{d}{dx}\bigg(\frac{1}{\text{cot}(x)}\bigg)$.

Answer: $\frac{1}{\text{sin}^2(x)}$. Chain rule on $\tan(x) = (\cot(x))^{-1}$ gives $\sec^2(x)$.

Flashcard 11: Evaluate $\frac{d}{dx}\bigg(\frac{1}{\text{cot}(x)}\bigg)$.

Answer: $\frac{1}{\text{sin}^2(x)}$. Chain rule on $\tan(x) = (\cot(x))^{-1}$ gives $\sec^2(x)$.

Flashcard 12: Evaluate $\frac{d}{dx}\bigg(\frac{1}{x^2 + 1}\bigg)$.

Answer: $-\frac{2x}{(x^2 + 1)^2}$. Chain rule on $(x^2 + 1)^{-1}$ gives $-1 \cdot (x^2 + 1)^{-2} \cdot 2x$.

Flashcard 13: Evaluate $\frac{d}{dx} \bigg(\frac{1}{x}\bigg)$.

Answer: $-\frac{1}{x^2}$. Power rule: $\frac{d}{dx}(x^{-1}) = -1 \cdot x^{-2}$.

Flashcard 14: Find the derivative of $e^{2x}$.

Answer: $2e^{2x}$. Chain rule: $\frac{d}{dx}(e^{2x}) = e^{2x} \cdot 2$.

Flashcard 15: Evaluate $\frac{d}{dx}\bigg(\frac{1}{x^2 + 1}\bigg)$.

Answer: $-\frac{2x}{(x^2 + 1)^2}$. Chain rule on $(x^2 + 1)^{-1}$ gives $-1 \cdot (x^2 + 1)^{-2} \cdot 2x$.

Flashcard 16: State the integral of $\frac{1}{x}$ with respect to $x$.

Answer: $\text{ln}|x| + C$. Standard antiderivative formula for $\frac{1}{x}$.

Flashcard 17: What is $\frac{d}{dx}\bigg(\frac{1}{x}\bigg)$?

Answer: $-\frac{1}{x^2}$. Derivative of $x^{-1}$ using power rule.

Flashcard 18: What is the integral of $\text{e}^{ax}$?

Answer: $\frac{1}{a}\text{e}^{ax} + C$. Standard exponential integral with constant $a$.

Flashcard 19: Evaluate $\frac{d}{dx}\bigg(\frac{1}{\text{cos}(x)}\bigg)$.

Answer: $\frac{\text{sin}(x)}{\text{cos}^2(x)}$. Chain rule on $\sec(x) = (\cos(x))^{-1}$ gives $\sec(x)\tan(x)$.

Flashcard 20: Evaluate $\frac{d}{dx}\bigg(\frac{1}{\tan(x)}\bigg)$.

Answer: $-\frac{1}{\sin^2(x)}$. Chain rule on $\cot(x) = (\tan(x))^{-1}$ gives $-\csc^2(x)$.

Flashcard 21: What is the integral of $\text{e}^{ax}$?

Answer: $\frac{1}{a}\text{e}^{ax} + C$. Standard exponential integral with constant $a$.

Flashcard 22: What is the integral of $x^n$ with respect to $x$?

Answer: $\frac{x^{n+1}}{n+1} + C$ for $n \neq -1$. Power rule for integration increases exponent by 1.

Flashcard 23: Evaluate $\frac{d}{dx}\bigg(\frac{1}{\text{e}^{ax}}\bigg)$.

Answer: $-a\text{e}^{-ax}$. Chain rule on $e^{-ax}$ gives $-a \cdot e^{-ax}$.

Flashcard 24: Evaluate $\frac{d}{dx}\bigg(\frac{1}{\text{sin}(x)}\bigg)$.

Answer: $-\frac{\text{cos}(x)}{\text{sin}^2(x)}$. Chain rule on $\csc(x) = (\sin(x))^{-1}$ gives $-\csc(x)\cot(x)$.

Flashcard 25: Evaluate $\frac{d}{dx}\bigg(\frac{1}{\text{csc}(x)}\bigg)$.

Answer: $\text{cot}(x)\text{csc}(x)$. Chain rule on $\sin(x) = (\csc(x))^{-1}$ gives $\cos(x)$.

Flashcard 26: What is the value of $\frac{d}{dx}\bigg(\frac{1}{x}\bigg)$?

Answer: $-\frac{1}{x^2}$. Power rule applied to $x^{-1}$.

Flashcard 27: Find $\frac{d}{dx} \bigg( \text{ln}|x| \bigg)$.

Answer: $\frac{1}{x}$. Derivative of natural logarithm is $\frac{1}{x}$.

Flashcard 28: Find the derivative of $e^{2x}$.

Answer: $2e^{2x}$. Chain rule: $\frac{d}{dx}(e^{2x}) = e^{2x} \cdot 2$.

Flashcard 29: Evaluate $\frac{d}{dx} \bigg(\frac{1}{x}\bigg)$.

Answer: $-\frac{1}{x^2}$. Power rule: $\frac{d}{dx}(x^{-1}) = -1 \cdot x^{-2}$.

Flashcard 30: Find $\frac{d}{dx} \bigg( \text{ln}|x| \bigg)$.

Answer: $\frac{1}{x}$. Derivative of natural logarithm is $\frac{1}{x}$.

Flashcard 31: Evaluate $\frac{d}{dx}\bigg(\frac{1}{\text{tan}(x)}\bigg)$.

Answer: $-\frac{1}{\text{sin}^2(x)}$. Chain rule on $\cot(x) = (\tan(x))^{-1}$ gives $-\csc^2(x)$.

Flashcard 32: What is the antiderivative of $e^x$?

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

Flashcard 33: What is $\frac{d}{dx}\bigg(\frac{1}{x}\bigg)$?

Answer: $-\frac{1}{x^2}$. Derivative of $x^{-1}$ using power rule.

Flashcard 34: Evaluate $\frac{d}{dx}\bigg(\frac{1}{\text{e}^{ax}}\bigg)$.

Answer: $-a\text{e}^{-ax}$. Chain rule on $e^{-ax}$ gives $-a \cdot e^{-ax}$.

Flashcard 35: Evaluate $\frac{d}{dx}\bigg(\frac{1}{\text{sin}(x)}\bigg)$.

Answer: $-\frac{\text{cos}(x)}{\text{sin}^2(x)}$. Chain rule on $\csc(x) = (\sin(x))^{-1}$ gives $-\csc(x)\cot(x)$.

Flashcard 36: State the integral of $\frac{1}{x}$ with respect to $x$.

Answer: $\ln |x| + C$. Standard antiderivative formula for $\frac{1}{x}$.