AP Calculus AB Flashcards: Selecting Techniques For Antidifferentiation

What this deck covers

This deck focuses on Selecting Techniques For Antidifferentiation, 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 antiderivative of $\text{cos}^2(x)$?

Answer: $\frac{x}{2} + \frac{1}{4}\text{sin}(2x) + C$. Use double angle identity: $\text{cos}^2(x) = \frac{1 + \text{cos}(2x)}{2}$.

Flashcard 2: Choose the technique for $\text{ln}(a^x)$.

Answer: Substitution. Simplify $\text{ln}(a^x) = x\text{ln}(a)$ before integrating.

Flashcard 3: Identify the technique for $e^{3x}\text{sin}(x)$.

Answer: Integration by parts. Product of exponential and sine requires integration by parts.

Flashcard 4: What is the antiderivative of $\cot(x)$?

Answer: $\ln |\sin(x)| + C$. Rewrite $\cot(x) = \frac{\cos(x)}{\sin(x)}$ and use substitution.

Flashcard 5: What is the antiderivative of $\text{cot}(x)$?

Answer: $\text{ln}|\text{sin}(x)| + C$. Rewrite $\text{cot}(x) = \frac{\text{cos}(x)}{\text{sin}(x)}$ and use substitution.

Flashcard 6: Find the antiderivative of $xe^{x}$.

Answer: Integration by parts. Product of polynomial and exponential requires integration by parts.

Flashcard 7: What is the antiderivative of $\frac{1}{\sqrt{1-x^2}}$?

Answer: $\arcsin(x) + C$. Standard inverse sine antiderivative formula.

Flashcard 8: What is the antiderivative of $\frac{1}{x}$?

Answer: $\text{ln}|x| + C$. The derivative of $\text{ln}|x|$ is $\frac{1}{x}$.

Flashcard 9: Which substitution is used for $\frac{1}{\text{sqrt}(a^2 - x^2)}$?

Answer: Trigonometric substitution. Use $x = a\text{sin}(\theta)$ to eliminate the square root.

Flashcard 10: Identify the technique for $\frac{1}{\text{sqrt}(x^2 + a^2)}$?

Answer: Trigonometric substitution. Use $x = a\text{tan}(\theta)$ to simplify the square root.

Flashcard 11: Identify the technique for $\text{arccos}(x)$.

Answer: Integration by parts. Inverse trig functions require integration by parts technique.

Flashcard 12: What is the antiderivative of $e^x$?

Answer: $e^x + C$. The derivative of $e^x$ is itself, so the antiderivative reverses this.

Flashcard 13: What is the antiderivative of $\frac{1}{x^2}$?

Answer: $-\frac{1}{x} + C$. Rewrite as $x^{-2}$ and apply power rule.

Flashcard 14: Find the antiderivative of $\text{sec}^2(x)$.

Answer: $\text{tan}(x) + C$. The derivative of $\text{tan}(x)$ is $\text{sec}^2(x)$.

Flashcard 15: Select the technique for $e^{x^2}$.

Answer: No elementary antiderivative. Cannot be expressed using elementary functions.

Flashcard 16: State the antiderivative of $\text{cos}(x)$.

Answer: $\text{sin}(x) + C$. The derivative of $\text{sin}(x)$ is $\text{cos}(x)$.

Flashcard 17: What is the antiderivative of $\frac{1}{\text{sqrt}(1-x^2)}$?

Answer: $\text{arcsin}(x) + C$. Standard inverse sine antiderivative formula.

Flashcard 18: What is the antiderivative of $a^x$?

Answer: $\frac{a^x}{\text{ln}(a)} + C$. General exponential function antiderivative formula with base $a$.

Flashcard 19: What is the antiderivative of $\frac{1}{x^2 + a^2}$?

Answer: $\frac{1}{a}\text{arctan}(\frac{x}{a}) + C$. Standard arctangent formula with scaling factor $\frac{1}{a}$.

Flashcard 20: What is the antiderivative of $\text{csc}(x)\text{cot}(x)$?

Answer: $-\text{csc}(x) + C$. The derivative of $-\text{csc}(x)$ is $\text{csc}(x)\text{cot}(x)$.

Flashcard 21: What is the antiderivative of $e^x$?

Answer: $e^x + C$. The derivative of $e^x$ is itself, so the antiderivative reverses this.

Flashcard 22: State the antiderivative of $\text{cos}(x)$.

Answer: $\text{sin}(x) + C$. The derivative of $\text{sin}(x)$ is $\text{cos}(x)$.

Flashcard 23: What substitution is used for $\text{sqrt}(x^2 + a^2)$?

Answer: Trigonometric substitution. Use $x = a\text{tan}(\theta)$ to handle the square root expression.

Flashcard 24: What is the antiderivative of $\text{sin}^2(x)$?

Answer: $\frac{x}{2} - \frac{1}{4}\text{sin}(2x) + C$. Use double angle identity: $\text{sin}^2(x) = \frac{1 - \text{cos}(2x)}{2}$.

Flashcard 25: What is the antiderivative of $\text{sec}(x)\text{tan}(x)$?

Answer: $\text{sec}(x) + C$. The derivative of $\text{sec}(x)$ is $\text{sec}(x)\text{tan}(x)$.

Flashcard 26: What is the antiderivative of $x^{n}$?

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

Flashcard 27: Which method is suitable for $x\text{cos}(x)$?

Answer: Integration by parts. Product of polynomial and cosine requires integration by parts.

Flashcard 28: Which technique is suitable for $\frac{x}{x^2 + 1}$?

Answer: Substitution. Use $u = x^2 + 1$ to simplify the integrand.

Flashcard 29: Identify the technique for $\text{arccos}(x)$.

Answer: Integration by parts. Inverse trig functions require integration by parts technique.

Flashcard 30: What is the antiderivative of $\text{csc}^2(x)$?

Answer: $-\text{cot}(x) + C$. The derivative of $-\text{cot}(x)$ is $\text{csc}^2(x)$.

Flashcard 31: What is the antiderivative of $\text{sin}^2(x)$?

Answer: $\frac{x}{2} - \frac{1}{4}\text{sin}(2x) + C$. Use double angle identity: $\text{sin}^2(x) = \frac{1 - \text{cos}(2x)}{2}$.

Flashcard 32: Choose the technique for $\text{ln}(a^x)$.

Answer: Substitution. Simplify $\text{ln}(a^x) = x\text{ln}(a)$ before integrating.

Flashcard 33: What substitution is used for $\text{sqrt}(x^2 + a^2)$?

Answer: Trigonometric substitution. Use $x = a\text{tan}(\theta)$ to handle the square root expression.

Flashcard 34: Which technique is suitable for $\frac{x}{x^2 + 1}$?

Answer: Substitution. Use $u = x^2 + 1$ to simplify the integrand.

Flashcard 35: What is the antiderivative of $x^{n}$?

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

Flashcard 36: What is the antiderivative of $\text{csc}(x)\text{cot}(x)$?

Answer: $-\text{csc}(x) + C$. The derivative of $-\text{csc}(x)$ is $\text{csc}(x)\text{cot}(x)$.

Flashcard 37: What is the antiderivative of $\text{arcsin}(x)$?

Answer: $x \text{arcsin}(x) + \text{sqrt}(1 - x^2) + C$. Standard result from integration by parts formula.

Flashcard 38: What is the antiderivative of $\sec(x)\tan(x)$?

Answer: $\sec(x) + C$. The derivative of $\sec(x)$ is $\sec(x)\tan(x)$.

Flashcard 39: Select the technique for $e^{x^2}$.

Answer: No elementary antiderivative. Cannot be expressed using elementary functions.

Flashcard 40: Identify the technique for $x^2 \text{ln}(x)$.

Answer: Integration by parts. Polynomial times logarithm requires integration by parts.

Flashcard 41: Choose the integration technique for $\text{arctan}(x)$.

Answer: Integration by parts. Inverse trig functions require integration by parts technique.

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

Answer: $\text{ln}|x| + C$. The derivative of $\text{ln}|x|$ is $\frac{1}{x}$.

Flashcard 43: What is the antiderivative of $\text{tan}(x)$?

Answer: $-\text{ln}|\text{cos}(x)| + C$. Rewrite $\text{tan}(x) = \frac{\text{sin}(x)}{\text{cos}(x)}$ and use substitution.

Flashcard 44: What is the antiderivative of $\text{cos}^2(x)$?

Answer: $\frac{x}{2} + \frac{1}{4}\text{sin}(2x) + C$. Use double angle identity: $\text{cos}^2(x) = \frac{1 + \text{cos}(2x)}{2}$.

Flashcard 45: What is the antiderivative of $\text{csc}^2(x)$?

Answer: $-\text{cot}(x) + C$. The derivative of $-\text{cot}(x)$ is $\text{csc}^2(x)$.

Flashcard 46: Identify the technique for $\frac{1}{\text{sqrt}(x^2 + a^2)}$?

Answer: Trigonometric substitution. Use $x = a\text{tan}(\theta)$ to simplify the square root.

Flashcard 47: What is the antiderivative of $\text{arcsin}(x)$?

Answer: $x \text{arcsin}(x) + \sqrt{1 - x^2} + C$. Standard result from integration by parts formula.

Flashcard 48: Choose the integration technique for $\text{arctan}(x)$.

Answer: Integration by parts. Inverse trig functions require integration by parts technique.

Flashcard 49: Which technique is appropriate for $e^{2x}$?

Answer: Substitution. Use $u = 2x$ substitution to handle the coefficient.

Flashcard 50: Identify the integration technique for $x\text{sin}(x)$.

Answer: Integration by parts. Product of $x$ and trig function requires integration by parts.

Flashcard 51: State the antiderivative of $\text{sin}(x)$.

Answer: $-\text{cos}(x) + C$. The derivative of $-\text{cos}(x)$ is $\text{sin}(x)$.

Flashcard 52: Identify the technique for $e^{3x}\text{sin}(x)$.

Answer: Integration by parts. Product of exponential and sine requires integration by parts.

Flashcard 53: What is the antiderivative of $\text{tan}(x)$?

Answer: $-\text{ln}|\text{cos}(x)| + C$. Rewrite $\text{tan}(x) = \frac{\text{sin}(x)}{\text{cos}(x)}$ and use substitution.

Flashcard 54: What is the antiderivative of $\frac{1}{x^2 + a^2}$?

Answer: $\frac{1}{a}\arctan(\frac{x}{a}) + C$. Standard arctangent formula with scaling factor $\frac{1}{a}$.

Flashcard 55: State the antiderivative of $\text{sin}(x)$.

Answer: $-\text{cos}(x) + C$. The derivative of $-\text{cos}(x)$ is $\text{sin}(x)$.

Flashcard 56: Identify the integration technique for $x\text{sin}(x)$.

Answer: Integration by parts. Product of $x$ and trig function requires integration by parts.

Flashcard 57: Find the antiderivative of $xe^{x}$.

Answer: Integration by parts. Product of polynomial and exponential requires integration by parts.

Flashcard 58: Which technique is appropriate for $e^{2x}$?

Answer: Substitution. Use $u = 2x$ substitution to handle the coefficient.

Flashcard 59: What is the antiderivative of $a^x$?

Answer: $\frac{a^x}{\ln(a)} + C$. General exponential function antiderivative formula with base $a$.

Flashcard 60: Identify the integration technique for $\text{ln}(x)$.

Answer: Integration by parts. Use $u = \text{ln}(x)$ and $dv = dx$ for integration by parts.

Flashcard 61: Find the antiderivative of $\text{sec}^2(x)$.

Answer: $\text{tan}(x) + C$. The derivative of $\text{tan}(x)$ is $\text{sec}^2(x)$.

Flashcard 62: Identify the technique for $x^2 \text{ln}(x)$.

Answer: Integration by parts. Polynomial times logarithm requires integration by parts.

Flashcard 63: Which substitution is used for $\frac{1}{\text{sqrt}(a^2 - x^2)}$?

Answer: Trigonometric substitution. Use $x = a\text{sin}(\theta)$ to eliminate the square root.

Flashcard 64: Identify the integration technique for $\text{ln}(x)$.

Answer: Integration by parts. Use $u = \text{ln}(x)$ and $dv = dx$ for integration by parts.

Flashcard 65: What is the antiderivative of $\frac{1}{x^2}$?

Answer: $-\frac{1}{x} + C$. Rewrite as $x^{-2}$ and apply power rule.

Flashcard 66: Which method is suitable for $x\text{cos}(x)$?

Answer: Integration by parts. Product of polynomial and cosine requires integration by parts.