AP Calculus AB Flashcards: Finding Antiderivatives And Indefinite Integrals

Study Finding Antiderivatives And Indefinite Integrals in AP Calculus AB with focused flashcards that help you recognize the idea, recall the key rule, and apply it in practice-style prompts.

AP Calculus AB

Finding Antiderivatives And Indefinite Integrals

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QUESTION

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What is the antiderivative of $f(x) = \sec^2(x)$ ?

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ANSWER

$F(x) = \tan(x) + C$ . Derivative of $\tan(x)$ is $\sec^2(x)$ , so reverse gives this.

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What this deck covers

This deck focuses on Finding Antiderivatives And Indefinite 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.

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Flashcard 1: What is the antiderivative of $f(x) = \sec^2(x)$ ?

Answer: $F(x) = \tan(x) + C$ . Derivative of $\tan(x)$ is $\sec^2(x)$ , so reverse gives this.

Flashcard 2: Identify the rule used to find the antiderivative of $cf(x)$ .

Answer: Constant multiple rule: $F(x) = cF(x) + C$ . Constants can be factored out of integrals.

Flashcard 3: Which rule is applied for $\frac{d}{dx} \text{F}(x) = f(x)$ ?

Answer: The Fundamental Theorem of Calculus, Part 1. States that antiderivative and derivative are inverse operations.

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

Answer: $F(x) = -\text{cos}(x) + C$ . Derivative of $-\cos(x)$ is $\sin(x)$ , so reverse gives this.

Flashcard 5: Identify the rule used to find the antiderivative of $cf(x)$ .

Answer: Constant multiple rule: $F(x) = cF(x) + C$ . Constants can be factored out of integrals.

Flashcard 6: Find the indefinite integral of $f(x) = 2x + 5$ .

Answer: $F(x) = x^2 + 5x + C$ . Apply power rule to each term: $\int 2x dx + \int 5 dx$ .

Flashcard 7: What is the indefinite integral of $f(x) = 6x^5 + 3x^2$ ?

Answer: $F(x) = x^6 + x^3 + C$ . Apply power rule to each term separately.

Flashcard 8: Find the indefinite integral of $f(x) = 7$ .

Answer: $F(x) = 7x + C$ . Antiderivative of constant $7$ is $7x$ .

Flashcard 9: State the notation for the indefinite integral of $f(x)$ .

Answer: $\text{F}(x) = \text{∫} f(x) \text{dx}$ . Standard integral notation with differential $dx$ .

Flashcard 10: What is the antiderivative of $f(x) = e^x$ ?

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

Flashcard 11: What is the antiderivative of $f(x) = \frac{1}{x}$ ?

Answer: $F(x) = \text{ln}|x| + C$ . Standard antiderivative of reciprocal function.

Flashcard 12: What is the antiderivative of $f(x) = x^n$ where $n \neq -1$ ?

Answer: $F(x) = \frac{x^{n+1}}{n+1} + C$ . Power rule: increase exponent by 1, divide by new exponent.

Flashcard 13: What is the indefinite integral of $f(x) = 8x^{-3}$ ?

Answer: $F(x) = -4x^{-2} + C$ . Apply power rule: $\int 8x^{-3} dx = \frac{8x^{-2}}{-2}$ .

Flashcard 14: What is the antiderivative of $f(x) = \text{cos}(x)$ ?

Answer: $F(x) = \text{sin}(x) + C$ . Derivative of $\sin(x)$ is $\cos(x)$ , so reverse gives this.

Flashcard 15: Find the indefinite integral of $f(x) = 7$ .

Answer: $F(x) = 7x + C$ . Antiderivative of constant $7$ is $7x$ .

Flashcard 16: Which rule is applied for $\frac{d}{dx} \text{F}(x) = f(x)$ ?

Answer: The Fundamental Theorem of Calculus, Part 1. States that antiderivative and derivative are inverse operations.

Flashcard 17: Find the indefinite integral of $f(x) = x^3 - 4x^2 + 6$ .

Answer: $F(x) = \frac{x^4}{4} - \frac{4x^3}{3} + 6x + C$ . Apply power rule to each term separately.

Flashcard 18: What is the antiderivative of $f(x) = \text{sec}^2(x)$ ?

Answer: $F(x) = \text{tan}(x) + C$ . Derivative of $\tan(x)$ is $\sec^2(x)$ , so reverse gives this.

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

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

Flashcard 20: What is the indefinite integral of $f(x) = \frac{1}{x^2}$ ?

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

Flashcard 21: What is the indefinite integral of $f(x) = -9x^5$ ?

Answer: $F(x) = -\frac{3}{2}x^6 + C$ . Apply power rule: $\int -9x^5 dx = \frac{-9x^6}{6}$ .

Flashcard 22: What is the indefinite integral of $f(x) = \frac{3}{x}$ ?

Answer: $F(x) = 3 \ln |x| + C$ . Constant multiple of $\frac{1}{x}$ antiderivative.

Flashcard 23: Find the indefinite integral of $f(x) = x^{-1/2}$ .

Answer: $F(x) = 2x^{1/2} + C$ . Apply power rule: $\int x^{-1/2} dx = \frac{x^{1/2}}{1/2} = 2x^{1/2}$ .

Flashcard 24: Find the indefinite integral of $f(x) = 3x^{-2}$ .

Answer: $F(x) = -\frac{3}{x} + C$ . Apply power rule: $\int 3x^{-2} dx = \frac{3x^{-1}}{-1} = -\frac{3}{x}$ .

Flashcard 25: Which rule is used to find the antiderivative of a sum $f(x) + g(x)$ ?

Answer: The sum rule: $\text{F}(x) = \text{F}_1(x) + \text{F}_2(x) + C$ . Antiderivative distributes over addition.

Flashcard 26: What is the antiderivative of $f(x) = \text{csc}^2(x)$ ?

Answer: $F(x) = -\text{cot}(x) + C$ . Derivative of $-\cot(x)$ is $\csc^2(x)$ , so reverse gives this.

Flashcard 27: Find the indefinite integral of $f(x) = 3x^{-2}$ .

Answer: $F(x) = -\frac{3}{x} + C$ . Apply power rule: $\int 3x^{-2} dx = \frac{3x^{-1}}{-1} = -\frac{3}{x}$ .

Flashcard 28: What is the indefinite integral of $f(x) = x^2 + 2x + 1$ ?

Answer: $F(x) = \frac{x^3}{3} + x^2 + x + C$ . Apply power rule to each term: perfect square trinomial.

Flashcard 29: What is the antiderivative of $f(x) = x^n$ where $n \neq -1$ ?

Answer: $F(x) = \frac{x^{n+1}}{n+1} + C$ . Power rule: increase exponent by 1, divide by new exponent.

Flashcard 30: What is the antiderivative of $f(x) = \text{sec}(x)\text{tan}(x)$ ?

Answer: $F(x) = \text{sec}(x) + C$ . Derivative of $\sec(x)$ is $\sec(x)\tan(x)$ , so reverse gives this.

Flashcard 31: What is the indefinite integral of $f(x) = 5x^{1/2}$ ?

Answer: $F(x) = \frac{10}{3}x^{3/2} + C$ . Apply power rule: $\int 5x^{1/2} dx = \frac{5x^{3/2}}{3/2}$ .

Flashcard 32: What is the indefinite integral of $f(x) = 8x^{-3}$ ?

Answer: $F(x) = -4x^{-2} + C$ . Apply power rule: $\int 8x^{-3} dx = \frac{8x^{-2}}{-2}$ .

Flashcard 33: Find the indefinite integral of $f(x) = 2x + 5$ .

Answer: $F(x) = x^2 + 5x + C$ . Apply power rule to each term: $\int 2x dx + \int 5 dx$ .

Flashcard 34: What is the antiderivative of $f(x) = \text{csc}(x)\text{cot}(x)$ ?

Answer: $F(x) = -\text{csc}(x) + C$ . Derivative of $-\csc(x)$ is $\csc(x)\cot(x)$ , so reverse gives this.

Flashcard 35: Find the indefinite integral of $f(x) = 4x^3 - 2x$ .

Answer: $F(x) = x^4 - x^2 + C$ . Apply power rule to each term separately.

Flashcard 36: Find the indefinite integral of $f(x) = 5x^4$ .

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

Flashcard 37: What is the indefinite integral of $f(x) = 6x^5 + 3x^2$ ?

Answer: $F(x) = x^6 + x^3 + C$ . Apply power rule to each term separately.

Flashcard 38: What is the antiderivative of $f(x) = \text{sec}(x)\text{tan}(x)$ ?

Answer: $F(x) = \text{sec}(x) + C$ . Derivative of $\sec(x)$ is $\sec(x)\tan(x)$ , so reverse gives this.

Flashcard 39: What is the indefinite integral of $f(x) = \frac{3}{x}$ ?

Answer: $F(x) = 3 \text{ln}|x| + C$ . Constant multiple of $\frac{1}{x}$ antiderivative.

Flashcard 40: Which rule is used to find the antiderivative of a sum $f(x) + g(x)$ ?

Answer: The sum rule: $\text{F}(x) = \text{F}_1(x) + \text{F}_2(x) + C$ . Antiderivative distributes over addition.

Flashcard 41: What is the antiderivative of $f(x) = \text{sin}(x)$ ?

Answer: $F(x) = -\text{cos}(x) + C$ . Derivative of $-\cos(x)$ is $\sin(x)$ , so reverse gives this.

Flashcard 42: What is the antiderivative of $f(x) = \text{csc}(x)\text{cot}(x)$ ?

Answer: $F(x) = -\text{csc}(x) + C$ . Derivative of $-\csc(x)$ is $\csc(x)\cot(x)$ , so reverse gives this.

Flashcard 43: Find the indefinite integral of $f(x) = x^{-1/2}$ .

Answer: $F(x) = 2x^{1/2} + C$ . Apply power rule: $\int x^{-1/2} dx = \frac{x^{1/2}}{1/2} = 2x^{1/2}$ .

Flashcard 44: Find the indefinite integral of $f(x) = \frac{4}{x^3}$ .

Answer: $F(x) = -2x^{-2} + C$ . Rewrite as $4x^{-3}$ and apply power rule.

Flashcard 45: What is the antiderivative of $f(x) = \frac{1}{x}$ ?

Answer: $F(x) = \text{ln}|x| + C$ . Standard antiderivative of reciprocal function.

Flashcard 46: What is the antiderivative of $f(x) = a$ (a constant)?

Answer: $F(x) = ax + C$ . Antiderivative of constant is constant times $x$ .

Flashcard 47: Find the indefinite integral of $f(x) = x^3 - 4x^2 + 6$ .

Answer: $F(x) = \frac{x^4}{4} - \frac{4x^3}{3} + 6x + C$ . Apply power rule to each term separately.

Flashcard 48: What is the indefinite integral of $f(x) = -9x^5$ ?

Answer: $F(x) = -\frac{3}{2}x^6 + C$ . Apply power rule: $\int -9x^5 dx = \frac{-9x^6}{6}$ .

Flashcard 49: What is the indefinite integral of $f(x) = 5x^{1/2}$ ?

Answer: $F(x) = \frac{10}{3}x^{3/2} + C$ . Apply power rule: $\int 5x^{1/2} dx = \frac{5x^{3/2}}{3/2}$ .

Flashcard 50: Find the indefinite integral of $f(x) = \frac{4}{x^3}$ .

Answer: $F(x) = -2x^{-2} + C$ . Rewrite as $4x^{-3}$ and apply power rule.

Flashcard 51: State the notation for the indefinite integral of $f(x)$ .

Answer: $\text{F}(x) = \text{∫} f(x) \text{dx}$ . Standard integral notation with differential $dx$ .

Flashcard 52: What is the antiderivative of $f(x) = \text{cos}(x)$ ?

Answer: $F(x) = \text{sin}(x) + C$ . Derivative of $\sin(x)$ is $\cos(x)$ , so reverse gives this.

Flashcard 53: Find the indefinite integral of $f(x) = 5x^4$ .

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

Flashcard 54: What is the indefinite integral of $f(x) = x^2 + 2x + 1$ ?

Answer: $F(x) = \frac{x^3}{3} + x^2 + x + C$ . Apply power rule to each term: perfect square trinomial.

Flashcard 55: What is the antiderivative of $f(x) = \text{csc}^2(x)$ ?

Answer: $F(x) = -\text{cot}(x) + C$ . Derivative of $-\cot(x)$ is $\csc^2(x)$ , so reverse gives this.

Flashcard 56: Find the indefinite integral of $f(x) = 4x^3 - 2x$ .

Answer: $F(x) = x^4 - x^2 + C$ . Apply power rule to each term separately.

Flashcard 57: What is the antiderivative of $f(x) = a$ (a constant)?

Answer: $F(x) = ax + C$ . Antiderivative of constant is constant times $x$ .

Flashcard 58: What is the indefinite integral of $f(x) = \frac{1}{x^2}$ ?

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

AP Calculus AB Flashcards: Finding Antiderivatives And Indefinite Integrals

Finding Antiderivatives And Indefinite Integrals

What this deck covers

How to use these flashcards

All flashcards

Flashcard 1: What is the antiderivative of f(x)=sec⁡2(x)f(x) = \sec^2(x)f(x)=sec2(x)?

Flashcard 2: Identify the rule used to find the antiderivative of cf(x)cf(x)cf(x).

Flashcard 3: Which rule is applied for ddxF(x)=f(x)\frac{d}{dx} \text{F}(x) = f(x)dxd​F(x)=f(x)?

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

Flashcard 5: Identify the rule used to find the antiderivative of cf(x)cf(x)cf(x).

Flashcard 6: Find the indefinite integral of f(x)=2x+5f(x) = 2x + 5f(x)=2x+5.

Flashcard 7: What is the indefinite integral of f(x)=6x5+3x2f(x) = 6x^5 + 3x^2f(x)=6x5+3x2?

Flashcard 8: Find the indefinite integral of f(x)=7f(x) = 7f(x)=7.

Flashcard 9: State the notation for the indefinite integral of f(x)f(x)f(x).

Flashcard 10: What is the antiderivative of f(x)=exf(x) = e^xf(x)=ex?

Flashcard 11: What is the antiderivative of f(x)=1xf(x) = \frac{1}{x}f(x)=x1​?

Flashcard 12: What is the antiderivative of f(x)=xnf(x) = x^nf(x)=xn where n≠−1n \neq -1n=−1?

Flashcard 13: What is the indefinite integral of f(x)=8x−3f(x) = 8x^{-3}f(x)=8x−3?

Flashcard 14: What is the antiderivative of f(x)=cos(x)f(x) = \text{cos}(x)f(x)=cos(x)?

Flashcard 15: Find the indefinite integral of f(x)=7f(x) = 7f(x)=7.

Flashcard 16: Which rule is applied for ddxF(x)=f(x)\frac{d}{dx} \text{F}(x) = f(x)dxd​F(x)=f(x)?

Flashcard 17: Find the indefinite integral of f(x)=x3−4x2+6f(x) = x^3 - 4x^2 + 6f(x)=x3−4x2+6.

Flashcard 18: What is the antiderivative of f(x)=sec2(x)f(x) = \text{sec}^2(x)f(x)=sec2(x)?

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

Flashcard 20: What is the indefinite integral of f(x)=1x2f(x) = \frac{1}{x^2}f(x)=x21​?

Flashcard 21: What is the indefinite integral of f(x)=−9x5f(x) = -9x^5f(x)=−9x5?

Flashcard 22: What is the indefinite integral of f(x)=3xf(x) = \frac{3}{x}f(x)=x3​?

Flashcard 23: Find the indefinite integral of f(x)=x−1/2f(x) = x^{-1/2}f(x)=x−1/2.

Flashcard 24: Find the indefinite integral of f(x)=3x−2f(x) = 3x^{-2}f(x)=3x−2.

Flashcard 25: Which rule is used to find the antiderivative of a sum f(x)+g(x)f(x) + g(x)f(x)+g(x)?

Flashcard 26: What is the antiderivative of f(x)=csc2(x)f(x) = \text{csc}^2(x)f(x)=csc2(x)?

Flashcard 27: Find the indefinite integral of f(x)=3x−2f(x) = 3x^{-2}f(x)=3x−2.

Flashcard 28: What is the indefinite integral of f(x)=x2+2x+1f(x) = x^2 + 2x + 1f(x)=x2+2x+1?

Flashcard 29: What is the antiderivative of f(x)=xnf(x) = x^nf(x)=xn where n≠−1n \neq -1n=−1?

Flashcard 30: What is the antiderivative of f(x)=sec(x)tan(x)f(x) = \text{sec}(x)\text{tan}(x)f(x)=sec(x)tan(x)?

Flashcard 31: What is the indefinite integral of f(x)=5x1/2f(x) = 5x^{1/2}f(x)=5x1/2?

Flashcard 32: What is the indefinite integral of f(x)=8x−3f(x) = 8x^{-3}f(x)=8x−3?

Flashcard 33: Find the indefinite integral of f(x)=2x+5f(x) = 2x + 5f(x)=2x+5.

Flashcard 34: What is the antiderivative of f(x)=csc(x)cot(x)f(x) = \text{csc}(x)\text{cot}(x)f(x)=csc(x)cot(x)?

Flashcard 35: Find the indefinite integral of f(x)=4x3−2xf(x) = 4x^3 - 2xf(x)=4x3−2x.

Flashcard 36: Find the indefinite integral of f(x)=5x4f(x) = 5x^4f(x)=5x4.

Flashcard 37: What is the indefinite integral of f(x)=6x5+3x2f(x) = 6x^5 + 3x^2f(x)=6x5+3x2?

Flashcard 38: What is the antiderivative of f(x)=sec(x)tan(x)f(x) = \text{sec}(x)\text{tan}(x)f(x)=sec(x)tan(x)?

Flashcard 39: What is the indefinite integral of f(x)=3xf(x) = \frac{3}{x}f(x)=x3​?

Flashcard 40: Which rule is used to find the antiderivative of a sum f(x)+g(x)f(x) + g(x)f(x)+g(x)?

Flashcard 41: What is the antiderivative of f(x)=sin(x)f(x) = \text{sin}(x)f(x)=sin(x)?

Flashcard 42: What is the antiderivative of f(x)=csc(x)cot(x)f(x) = \text{csc}(x)\text{cot}(x)f(x)=csc(x)cot(x)?

Flashcard 43: Find the indefinite integral of f(x)=x−1/2f(x) = x^{-1/2}f(x)=x−1/2.

Flashcard 44: Find the indefinite integral of f(x)=4x3f(x) = \frac{4}{x^3}f(x)=x34​.

Flashcard 45: What is the antiderivative of f(x)=1xf(x) = \frac{1}{x}f(x)=x1​?

Flashcard 46: What is the antiderivative of f(x)=af(x) = af(x)=a (a constant)?

Flashcard 47: Find the indefinite integral of f(x)=x3−4x2+6f(x) = x^3 - 4x^2 + 6f(x)=x3−4x2+6.

Flashcard 48: What is the indefinite integral of f(x)=−9x5f(x) = -9x^5f(x)=−9x5?

Flashcard 49: What is the indefinite integral of f(x)=5x1/2f(x) = 5x^{1/2}f(x)=5x1/2?

Flashcard 50: Find the indefinite integral of f(x)=4x3f(x) = \frac{4}{x^3}f(x)=x34​.

Flashcard 51: State the notation for the indefinite integral of f(x)f(x)f(x).

Flashcard 52: What is the antiderivative of f(x)=cos(x)f(x) = \text{cos}(x)f(x)=cos(x)?

Flashcard 53: Find the indefinite integral of f(x)=5x4f(x) = 5x^4f(x)=5x4.

Flashcard 54: What is the indefinite integral of f(x)=x2+2x+1f(x) = x^2 + 2x + 1f(x)=x2+2x+1?

Flashcard 55: What is the antiderivative of f(x)=csc2(x)f(x) = \text{csc}^2(x)f(x)=csc2(x)?

Flashcard 56: Find the indefinite integral of f(x)=4x3−2xf(x) = 4x^3 - 2xf(x)=4x3−2x.

Flashcard 57: What is the antiderivative of f(x)=af(x) = af(x)=a (a constant)?

Flashcard 58: What is the indefinite integral of f(x)=1x2f(x) = \frac{1}{x^2}f(x)=x21​?