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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.

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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.

AP Calculus AB Flashcards: Finding Antiderivatives And Indefinite Integrals

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QUESTION

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

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ANSWER

F(x)=7x+CF(x) = 7x + CF(x)=7x+C. Antiderivative of constant 777 is 7x7x7x.

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Flashcard 1: Find the indefinite integral of f(x)=7f(x) = 7f(x)=7.

Answer: F(x)=7x+CF(x) = 7x + CF(x)=7x+C. Antiderivative of constant 777 is 7x7x7x.

Flashcard 2: What is the indefinite integral of f(x)=−9x5f(x) = -9x^5f(x)=−9x5?

Answer: F(x)=−32x6+CF(x) = -\frac{3}{2}x^6 + CF(x)=−23​x6+C. Apply power rule: ∫−9x5dx=−9x66\int -9x^5 dx = \frac{-9x^6}{6}∫−9x5dx=6−9x6​.

Flashcard 3: Find the indefinite integral of f(x)=x−1/2f(x) = x^{-1/2}f(x)=x−1/2.

Answer: F(x)=2x1/2+CF(x) = 2x^{1/2} + CF(x)=2x1/2+C. Apply power rule: ∫x−1/2dx=x1/21/2=2x1/2\int x^{-1/2} dx = \frac{x^{1/2}}{1/2} = 2x^{1/2}∫x−1/2dx=1/2x1/2​=2x1/2.

Flashcard 4: What is the antiderivative of f(x)=csc2(x)f(x) = \text{csc}^2(x)f(x)=csc2(x)?

Answer: F(x)=−cot(x)+CF(x) = -\text{cot}(x) + CF(x)=−cot(x)+C. Derivative of −cot⁡(x)-\cot(x)−cot(x) is csc⁡2(x)\csc^2(x)csc2(x), so reverse gives this.

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

Answer: F(x)=x5+CF(x) = x^5 + CF(x)=x5+C. Apply power rule: ∫5x4dx=5x55=x5\int 5x^4 dx = \frac{5x^5}{5} = x^5∫5x4dx=55x5​=x5.

Flashcard 6: What is the antiderivative of f(x)=cos(x)f(x) = \text{cos}(x)f(x)=cos(x)?

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

Flashcard 7: Find the indefinite integral of f(x)=x3−4x2+6f(x) = x^3 - 4x^2 + 6f(x)=x3−4x2+6.

Answer: F(x)=x44−4x33+6x+CF(x) = \frac{x^4}{4} - \frac{4x^3}{3} + 6x + CF(x)=4x4​−34x3​+6x+C. Apply power rule to each term separately.

Flashcard 8: What is the indefinite integral of f(x)=x2+2x+1f(x) = x^2 + 2x + 1f(x)=x2+2x+1?

Answer: F(x)=x33+x2+x+CF(x) = \frac{x^3}{3} + x^2 + x + CF(x)=3x3​+x2+x+C. Apply power rule to each term: perfect square trinomial.

Flashcard 9: Which rule is used to find the antiderivative of a sum f(x)+g(x)f(x) + g(x)f(x)+g(x)?

Answer: The sum rule: F(x)=F1(x)+F2(x)+C\text{F}(x) = \text{F}_1(x) + \text{F}_2(x) + CF(x)=F1​(x)+F2​(x)+C. Antiderivative distributes over addition.

Flashcard 10: What is the indefinite integral of f(x)=8x−3f(x) = 8x^{-3}f(x)=8x−3?

Answer: F(x)=−4x−2+CF(x) = -4x^{-2} + CF(x)=−4x−2+C. Apply power rule: ∫8x−3dx=8x−2−2\int 8x^{-3} dx = \frac{8x^{-2}}{-2}∫8x−3dx=−28x−2​.

Flashcard 11: What is the antiderivative of f(x)=exf(x) = e^xf(x)=ex?

Answer: F(x)=ex+CF(x) = e^x + CF(x)=ex+C. Exponential function exe^xex is its own antiderivative.

Flashcard 12: What is the antiderivative of f(x)=af(x) = af(x)=a (a constant)?

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

Flashcard 13: What is the antiderivative of f(x)=sin(x)f(x) = \text{sin}(x)f(x)=sin(x)?

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

Flashcard 14: Identify the rule used to find the antiderivative of cf(x)cf(x)cf(x).

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

Flashcard 15: Find the indefinite integral of f(x)=4x3f(x) = \frac{4}{x^3}f(x)=x34​.

Answer: F(x)=−2x−2+CF(x) = -2x^{-2} + CF(x)=−2x−2+C. Rewrite as 4x−34x^{-3}4x−3 and apply power rule.

Flashcard 16: What is the indefinite integral of f(x)=6x5+3x2f(x) = 6x^5 + 3x^2f(x)=6x5+3x2?

Answer: F(x)=x6+x3+CF(x) = x^6 + x^3 + CF(x)=x6+x3+C. Apply power rule to each term separately.

Flashcard 17: What is the indefinite integral of f(x)=5x1/2f(x) = 5x^{1/2}f(x)=5x1/2?

Answer: F(x)=103x3/2+CF(x) = \frac{10}{3}x^{3/2} + CF(x)=310​x3/2+C. Apply power rule: ∫5x1/2dx=5x3/23/2\int 5x^{1/2} dx = \frac{5x^{3/2}}{3/2}∫5x1/2dx=3/25x3/2​.

Flashcard 18: Find the indefinite integral of f(x)=2x+5f(x) = 2x + 5f(x)=2x+5.

Answer: F(x)=x2+5x+CF(x) = x^2 + 5x + CF(x)=x2+5x+C. Apply power rule to each term: ∫2xdx+∫5dx\int 2x dx + \int 5 dx∫2xdx+∫5dx.

Flashcard 19: Find the indefinite integral of f(x)=4x3−2xf(x) = 4x^3 - 2xf(x)=4x3−2x.

Answer: F(x)=x4−x2+CF(x) = x^4 - x^2 + CF(x)=x4−x2+C. Apply power rule to each term separately.

Flashcard 20: Find the indefinite integral of f(x)=3x−2f(x) = 3x^{-2}f(x)=3x−2.

Answer: F(x)=−3x+CF(x) = -\frac{3}{x} + CF(x)=−x3​+C. Apply power rule: ∫3x−2dx=3x−1−1=−3x\int 3x^{-2} dx = \frac{3x^{-1}}{-1} = -\frac{3}{x}∫3x−2dx=−13x−1​=−x3​.

Flashcard 21: Which rule is applied for ddxF(x)=f(x)\frac{d}{dx} \text{F}(x) = f(x)dxd​F(x)=f(x)?

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

Flashcard 22: 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)?

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

Flashcard 23: 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)?

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

Flashcard 24: What is the antiderivative of f(x)=sec2(x)f(x) = \text{sec}^2(x)f(x)=sec2(x)?

Answer: F(x)=tan(x)+CF(x) = \text{tan}(x) + CF(x)=tan(x)+C. Derivative of tan⁡(x)\tan(x)tan(x) is sec⁡2(x)\sec^2(x)sec2(x), so reverse gives this.

Flashcard 25: What is the antiderivative of f(x)=1xf(x) = \frac{1}{x}f(x)=x1​?

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

Flashcard 26: What is the antiderivative of f(x)=xnf(x) = x^nf(x)=xn where n≠−1n \neq -1n=−1?

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

Flashcard 27: What is the indefinite integral of f(x)=1x2f(x) = \frac{1}{x^2}f(x)=x21​?

Answer: F(x)=−1x+CF(x) = -\frac{1}{x} + CF(x)=−x1​+C. Rewrite as x−2x^{-2}x−2 and apply power rule.

Flashcard 28: State the notation for the indefinite integral of f(x)f(x)f(x).

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

Flashcard 29: What is the indefinite integral of f(x)=3xf(x) = \frac{3}{x}f(x)=x3​?

Answer: F(x)=3ln⁡∣x∣+CF(x) = 3 \ln |x| + CF(x)=3ln∣x∣+C. Constant multiple of 1x\frac{1}{x}x1​ antiderivative.

Flashcard 30: What is the antiderivative of f(x)=af(x) = af(x)=a (a constant)?

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