AP Calculus BC Flashcards: Evaluating Improper Integrals

Study Evaluating Improper Integrals in AP Calculus BC with focused flashcards that help you recognize the idea, recall the key rule, and apply it in practice-style prompts.

AP Calculus BC

Evaluating Improper Integrals

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QUESTION

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Determine convergence: $\int_{0}^{\infty} e^{-x^2} \, dx$ .

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ANSWER

Converges. Gaussian integral converges due to exponential decay.

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What this deck covers

This deck focuses on Evaluating Improper Integrals, 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.

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Flashcard 1: Determine convergence: $\int_{0}^{\infty} e^{-x^2} \, dx$ .

Answer: Converges. Gaussian integral converges due to exponential decay.

Flashcard 2: When does an integral $\textstyle \int_{a}^{b} f(x) \, dx$ become improper?

Answer: If $a$ , $b$ , or $f(x)$ are infinite or unbounded. Any infinite bound or unbounded integrand makes it improper.

Flashcard 3: What is an improper integral?

Answer: An integral with infinite limits or an unbounded integrand. Defined when limits of integration or integrand are infinite.

Flashcard 4: Evaluate: $\textstyle \text{∫}_{0}^{1} \frac{1}{\text{√}x} \text{dx}$ .

Answer: Converges to $2$ . $\lim_{a \to 0^+} [2\sqrt{x}]_a^1 = 2\sqrt{1} - 0 = 2$ .

Flashcard 5: What does it mean if an improper integral converges?

Answer: The limit defining the integral exists and is finite. The limit exists and equals a finite real number.

Flashcard 6: Identify convergence: $\int_{1}^{\infty} \frac{1}{\sqrt{x}} \, dx$ .

Answer: Diverges. Since $p = \frac{1}{2} \leq 1$ , the integral diverges.

Flashcard 7: What does it mean if an improper integral converges?

Answer: The limit defining the integral exists and is finite. The limit exists and equals a finite real number.

Flashcard 8: Identify the singularity: $\textstyle \int_{0}^{1} \frac{1}{x} dx$ .

Answer: Unbounded integrand at $x = 0$ . Integrand has vertical asymptote at the lower limit.

Flashcard 9: Determine behavior: $\textstyle \text{∫}_{0}^{1} \frac{1}{\text{√}x} \text{dx}$

Answer: Converges. Power $-\frac{1}{2} > -1$ ensures convergence at $x = 0$ .

Flashcard 10: Evaluate: $\textstyle \text{∫}_{0}^{\text{∞}} e^{-x} \text{dx}$ .

Answer: Converges to $1$ . $\lim_{b \to \infty} [-e^{-x}]_0^b = 0 - (-1) = 1$ .

Flashcard 11: Convert to limit form: $\textstyle ∫_{-∞}^{0} e^{x} \, dx$ .

Answer: $\textstyle \lim_{a \to -∞} ∫_{a}^{0} e^{x} \, dx$ . Negative infinite lower limit requires limit form.

Flashcard 12: Convert to limit: $\textstyle \text{∫}_{0}^{1} \frac{1}{x} \text{dx}$ .

Answer: $\textstyle \text{lim}_{a \to 0^+} \text{∫}_{a}^{1} \frac{1}{x} \text{dx}$ . Integrand $\frac{1}{x}$ is unbounded at lower limit $x = 0$ .

Flashcard 13: Identify singularity: $\int_{1}^{\infty} \frac{1}{(x-1)^2} dx$ .

Answer: Unbounded integrand at $x=1$ . Function becomes infinite at the lower limit $x = 1$ .

Flashcard 14: Evaluate: $\textstyle \int_{1}^{\infty} e^{-x} \text{dx}$

Answer: Converges to $\frac{1}{e}$ . $\lim_{b \to \infty} [-e^{-x}]_1^b = 0 - (-e^{-1}) = \frac{1}{e}$

Flashcard 15: Evaluate: $\int_{0}^{1} \frac{1}{\sqrt{1-x^2}} \, dx$ .

Answer: Converges to $\frac{\pi}{2}$ . Arcsine function evaluation: $\arcsin(1) - \arcsin(0) = \frac{\pi}{2}$ .

Flashcard 16: State the condition for the convergence of $\int_{0}^{a} (a-x)^{-1/2} \, dx$ .

Answer: Converges if $a > 0$ . For $a > 0$ , the singularity at $x = a$ is integrable.

Flashcard 17: Evaluate: $\textstyle \int_{1}^{\infty} \frac{1}{x (\ln x)^2} \, dx$ .

Answer: Converges. Higher power in denominator ensures convergence.

Flashcard 18: What is convergence for $\textstyle \text{∫}_{1}^{\text{∞}} \frac{1}{x^4} \text{dx}$ ?

Answer: Converges to $\frac{1}{3}$ . $\lim_{b \to \infty} [-\frac{1}{3x^3}]_1^b = 0 - (-\frac{1}{3}) = \frac{1}{3}$ .

Flashcard 19: What is an improper integral?

Answer: An integral with infinite limits or an unbounded integrand. Defined when limits of integration or integrand are infinite.

Flashcard 20: What is convergence for $\int_1^\infty \frac{1}{x^4} \, dx$ ?

Answer: Converges to $\frac{1}{3}$ . $\lim_{b \to \infty} [-\frac{1}{3x^3}]_1^b = 0 - (-\frac{1}{3}) = \frac{1}{3}$ .

Flashcard 21: Evaluate limit: $\textstyle \lim_{a \to 0^+} \int_{a}^{1} \frac{1}{x} dx$ .

Answer: Diverges. $\lim_{a \to 0^+} [\ln x]_a^1 = 0 - (-\infty) = \infty$ .

Flashcard 22: Determine behavior: $\textstyle \int_{0}^{1} \frac{1}{x^p} \text{dx}$ for $p<1$ .

Answer: Converges. When $p < 1$ , the singularity at $x = 0$ is integrable.

Flashcard 23: Find the limit: $\textstyle \lim_{b \to \infty} \int_{1}^{b} \frac{1}{x^3} dx$ .

Answer: Converges to $\frac{1}{2}$ . $\lim_{b \to \infty} [-\frac{1}{2x^2}]_1^b = 0 - (-\frac{1}{2}) = \frac{1}{2}$ .

Flashcard 24: Evaluate: $\int_{1}^{\infty} e^{-x} \, dx$ .

Answer: Converges to $\frac{1}{e}$ . $\lim_{b \to \infty} [-e^{-x}]_1^b = 0 - (-e^{-1}) = \frac{1}{e}$

Flashcard 25: Evaluate: $\textstyle \text{∫}_{0}^{\text{∞}} e^{-x} \text{dx}$ .

Answer: Converges to $1$ . $\lim_{b \to \infty} [-e^{-x}]_0^b = 0 - (-1) = 1$ .

Flashcard 26: Find the limit: $\textstyle \lim_{b \to \infty} \int_{1}^{b} \frac{1}{x^3} \text{dx}$

Answer: Converges to $\frac{1}{2}$ . $\lim_{b \to \infty} [-\frac{1}{2x^2}]_1^b = 0 - (-\frac{1}{2}) = \frac{1}{2}$

Flashcard 27: Convert to limit: $\textstyle \int_{0}^{\infty} x e^{-x} \, dx$ .

Answer: $\textstyle \lim_{b \to \infty} \int_{0}^{b} x e^{-x} \, dx$ . Infinite upper limit requires limit evaluation.

Flashcard 28: Evaluate: $\int_{1}^{\infty} \frac{1}{x^2} dx$

Answer: Converges to $1$ . $\lim_{b \to \infty} [-\frac{1}{x}]_1^b = \lim_{b \to \infty} (0 + 1) = 1$

Flashcard 29: $\textstyle \int_{0}^{\infty} x e^{-x} \, dx$

Answer: $\textstyle \lim_{b \to \infty} \int_{0}^{b} x e^{-x} \, dx$ . Infinite upper limit requires limit evaluation.

Flashcard 30: Determine convergence: $\textstyle \int_{0}^{\infty} e^{-x^2} \text{dx}$

Answer: Converges. Gaussian integral converges due to exponential decay.

Flashcard 31: What is the behavior of $\textstyle \text{∫}_{0}^{1} x^{-1/3} \text{dx}$ ?

Answer: Converges. Power $-\frac{1}{3} > -1$ ensures convergence.

Flashcard 32: Identify the type of improper integral: $\int_{0}^{1} \frac{1}{\sqrt{x}} \text{dx}$ .

Answer: Unbounded integrand at $x = 0$ . $\frac{1}{\sqrt{x}}$ becomes infinite as $x \to 0^+.$

Flashcard 33: State the behavior of $\int_{0}^{1} x^{-1} dx$ .

Answer: Diverges. Same as $\int_0^1 \frac{1}{x} dx$ which diverges.

Flashcard 34: Identify singularity: $\textstyle \int_{1}^{\infty} \frac{1}{(x-1)^2} \text{d}x$ .

Answer: Unbounded integrand at $x=1$ . Function becomes infinite at the lower limit $x = 1$ .

Flashcard 35: Determine behavior: $\int_{0}^{1} \frac{1}{\sqrt{x}} \mathrm{d}x$ .

Answer: Converges. Power $-\frac{1}{2} > -1$ ensures convergence at $x = 0$ .

Flashcard 36: Evaluate: $\textstyle \int_{1}^{\infty} \frac{1}{x (\ln x)^2} \, dx$ .

Answer: Converges. Higher power in denominator ensures convergence.

Flashcard 37: What is the behavior of $\int_{-\infty}^{\infty} e^{-x^2} \, dx$ ?

Answer: Converges to $\sqrt{\pi}$ . Famous Gaussian integral equals $\sqrt{\pi}$ .

Flashcard 38: Identify convergence: $\int_{1}^{\infty} \frac{1}{\sqrt{x}} dx$ .

Answer: Diverges. Since $p = \frac{1}{2} \leq 1$ , the integral diverges.

Flashcard 39: Convert to limit form: $\int_{-\infty}^{0} e^{x} \, dx$ .

Answer: $\lim_{a \to -\infty} \int_{a}^{0} e^{x} \, dx$ . Negative infinite lower limit requires limit form.

Flashcard 40: What is the behavior of $\int_{-\infty}^{\infty} e^{-x^2} \, dx$ ?

Answer: Converges to $\sqrt{\pi}$ . Famous Gaussian integral equals $\sqrt{\pi}$ .

Flashcard 41: When does an integral $\int_{a}^{b} f(x) \, dx$ become improper?

Answer: If $a$ , $b$ , or $f(x)$ are infinite or unbounded. Any infinite bound or unbounded integrand makes it improper.

Flashcard 42: What is the behavior of $\textstyle \text{∫}_{0}^{1} x^{-1/3} \text{dx}$ ?

Answer: Converges. Power $-\frac{1}{3} > -1$ ensures convergence.

Flashcard 43: Evaluate limit: $\textstyle \lim_{a \to 0^+} \int_{a}^{1} \frac{1}{x} \text{dx}$

Answer: Diverges. $\lim_{a \to 0^+} [\ln x]_a^1 = 0 - (-\infty) = \infty$

Flashcard 44: Identify the singularity: $\textstyle \int_{0}^{1} \frac{1}{x} \, dx$ .

Answer: Unbounded integrand at $x = 0$ . Integrand has vertical asymptote at the lower limit.

Flashcard 45: What is the behavior of $\int_{-\infty}^{0} e^x \, dx$ ?

Answer: Converges to $1$ . $\lim_{a \to -\infty} [e^x]_a^0 = 1 - 0 = 1$ .

Flashcard 46: Explain divergence in the context of improper integrals.

Answer: The integral's limit is infinite or does not exist. The limit is infinite or undefined.

Flashcard 47: Explain divergence in the context of improper integrals.

Answer: The integral's limit is infinite or does not exist. The limit is infinite or undefined.

Flashcard 48: Convert to limit: $\int_{0}^{1} \frac{1}{x} \, \mathrm{d}x$ .

Answer: $\lim_{a \to 0^+} \int_{a}^{1} \frac{1}{x} \, \mathrm{d}x$ . Integrand $\frac{1}{x}$ is unbounded at lower limit $x = 0$ .

Flashcard 49: What is the behavior of $\textstyle \text{∫}_{-\text{∞}}^{0} e^x \text{dx}$ ?

Answer: Converges to $1$ . $\lim_{a \to -\infty} [e^x]_a^0 = 1 - 0 = 1$ .

Flashcard 50: State the condition for the convergence of $\textstyle \int_{0}^{a} (a-x)^{-1/2} \, dx$ .

Answer: Converges if $a > 0$ . For $a > 0$ , the singularity at $x = a$ is integrable.

Flashcard 51: Determine behavior: $\textstyle \int_{0}^{1} \frac{1}{x^p} dx$ for $p<1$ .

Answer: Converges. When $p < 1$ , the singularity at $x = 0$ is integrable.

Flashcard 52: Evaluate: $\textstyle \text{∫}_{0}^{1} \frac{1}{\text{√}x} \text{dx}$ .

Answer: Converges to $2$ . $\lim_{a \to 0^+} [2\sqrt{x}]_a^1 = 2\sqrt{1} - 0 = 2$ .

Flashcard 53: Evaluate: $\textstyle \int_{0}^{1} \frac{1}{\sqrt{1-x^2}} \, dx$ .

Answer: Converges to $\frac{\pi}{2}$ . Arcsine function evaluation: $\arcsin(1) - \arcsin(0) = \frac{\pi}{2}$ .

Flashcard 54: Evaluate: $\textstyle \text{∫}_{1}^{\text{∞}} \frac{1}{x^2} \text{dx}$ .

Answer: Converges to $1$ . $\lim_{b \to \infty} [-\frac{1}{x}]_1^b = \lim_{b \to \infty} (0 + 1) = 1$ .

Flashcard 55: State the behavior of $\int_0^1 x^{-1} \text{dx}$

Answer: Diverges. Same as $\int_0^1 \frac{1}{x} dx$ which diverges.

Flashcard 56: Determine convergence: $\textstyle \int_{1}^{\infty} \frac{1}{x} \text{dx}$ .

Answer: Diverges. $\ln(x)$ grows without bound, so the integral diverges.

AP Calculus BC Flashcards: Evaluating Improper Integrals

Evaluating Improper Integrals

What this deck covers

How to use these flashcards

All flashcards

Flashcard 1: Determine convergence: ∫0∞e−x2 dx\int_{0}^{\infty} e^{-x^2} \, dx∫0∞​e−x2dx.

Flashcard 2: When does an integral ∫abf(x) dx\textstyle \int_{a}^{b} f(x) \, dx∫ab​f(x)dx become improper?

Flashcard 3: What is an improper integral?

Flashcard 4: Evaluate: \textstyle \text{∫}_{0}^{1} \frac{1}{\text{√}x} \text{dx}.

Flashcard 5: What does it mean if an improper integral converges?

Flashcard 6: Identify convergence: ∫1∞1x dx\int_{1}^{\infty} \frac{1}{\sqrt{x}} \, dx∫1∞​x​1​dx.

Flashcard 7: What does it mean if an improper integral converges?

Flashcard 8: Identify the singularity: ∫011xdx\textstyle \int_{0}^{1} \frac{1}{x} dx∫01​x1​dx.

Flashcard 9: Determine behavior: \textstyle \text{∫}_{0}^{1} \frac{1}{\text{√}x} \text{dx}

Flashcard 10: Evaluate: \textstyle \text{∫}_{0}^{\text{∞}} e^{-x} \text{dx}.

Flashcard 11: Convert to limit form: ∫−∞0ex dx\textstyle ∫_{-∞}^{0} e^{x} \, dx∫−∞0​exdx.

Flashcard 12: Convert to limit: \textstyle \text{∫}_{0}^{1} \frac{1}{x} \text{dx}.

Flashcard 13: Identify singularity: ∫1∞1(x−1)2dx\int_{1}^{\infty} \frac{1}{(x-1)^2} dx∫1∞​(x−1)21​dx.

Flashcard 14: Evaluate: ∫1∞e−xdx\textstyle \int_{1}^{\infty} e^{-x} \text{dx}∫1∞​e−xdx

Flashcard 15: Evaluate: ∫0111−x2 dx\int_{0}^{1} \frac{1}{\sqrt{1-x^2}} \, dx∫01​1−x2​1​dx.

Flashcard 16: State the condition for the convergence of ∫0a(a−x)−1/2 dx\int_{0}^{a} (a-x)^{-1/2} \, dx∫0a​(a−x)−1/2dx.

Flashcard 17: Evaluate: ∫1∞1x(ln⁡x)2 dx\textstyle \int_{1}^{\infty} \frac{1}{x (\ln x)^2} \, dx∫1∞​x(lnx)21​dx.

Flashcard 18: What is convergence for \textstyle \text{∫}_{1}^{\text{∞}} \frac{1}{x^4} \text{dx}?

Flashcard 19: What is an improper integral?

Flashcard 20: What is convergence for ∫1∞1x4 dx\int_1^\infty \frac{1}{x^4} \, dx∫1∞​x41​dx?

Flashcard 21: Evaluate limit: lim⁡a→0+∫a11xdx\textstyle \lim_{a \to 0^+} \int_{a}^{1} \frac{1}{x} dxlima→0+​∫a1​x1​dx.

Flashcard 22: Determine behavior: ∫011xpdx\textstyle \int_{0}^{1} \frac{1}{x^p} \text{dx}∫01​xp1​dx for p<1p<1p<1.

Flashcard 23: Find the limit: lim⁡b→∞∫1b1x3dx\textstyle \lim_{b \to \infty} \int_{1}^{b} \frac{1}{x^3} dxlimb→∞​∫1b​x31​dx.

Flashcard 24: Evaluate: ∫1∞e−x dx\int_{1}^{\infty} e^{-x} \, dx∫1∞​e−xdx.

Flashcard 25: Evaluate: \textstyle \text{∫}_{0}^{\text{∞}} e^{-x} \text{dx}.

Flashcard 26: Find the limit: lim⁡b→∞∫1b1x3dx\textstyle \lim_{b \to \infty} \int_{1}^{b} \frac{1}{x^3} \text{dx}limb→∞​∫1b​x31​dx

Flashcard 27: Convert to limit: ∫0∞xe−x dx\textstyle \int_{0}^{\infty} x e^{-x} \, dx∫0∞​xe−xdx.

Flashcard 28: Evaluate: ∫1∞1x2dx\int_{1}^{\infty} \frac{1}{x^2} dx∫1∞​x21​dx

Flashcard 29: ∫0∞xe−x dx\textstyle \int_{0}^{\infty} x e^{-x} \, dx∫0∞​xe−xdx

Flashcard 30: Determine convergence: ∫0∞e−x2dx\textstyle \int_{0}^{\infty} e^{-x^2} \text{dx}∫0∞​e−x2dx

Flashcard 31: What is the behavior of \textstyle \text{∫}_{0}^{1} x^{-1/3} \text{dx}?

Flashcard 32: Identify the type of improper integral: ∫011xdx\int_{0}^{1} \frac{1}{\sqrt{x}} \text{dx}∫01​x​1​dx.

Flashcard 33: State the behavior of ∫01x−1dx\int_{0}^{1} x^{-1} dx∫01​x−1dx.

Flashcard 34: Identify singularity: ∫1∞1(x−1)2dx\textstyle \int_{1}^{\infty} \frac{1}{(x-1)^2} \text{d}x∫1∞​(x−1)21​dx.

Flashcard 35: Determine behavior: ∫011xdx\int_{0}^{1} \frac{1}{\sqrt{x}} \mathrm{d}x∫01​x​1​dx.

Flashcard 36: Evaluate: ∫1∞1x(ln⁡x)2 dx\textstyle \int_{1}^{\infty} \frac{1}{x (\ln x)^2} \, dx∫1∞​x(lnx)21​dx.

Flashcard 37: What is the behavior of ∫−∞∞e−x2 dx\int_{-\infty}^{\infty} e^{-x^2} \, dx∫−∞∞​e−x2dx?

Flashcard 38: Identify convergence: ∫1∞1xdx\int_{1}^{\infty} \frac{1}{\sqrt{x}} dx∫1∞​x​1​dx.

Flashcard 39: Convert to limit form: ∫−∞0ex dx\int_{-\infty}^{0} e^{x} \, dx∫−∞0​exdx.

Flashcard 40: What is the behavior of ∫−∞∞e−x2 dx\int_{-\infty}^{\infty} e^{-x^2} \, dx∫−∞∞​e−x2dx?

Flashcard 41: When does an integral ∫abf(x) dx\int_{a}^{b} f(x) \, dx∫ab​f(x)dx become improper?

Flashcard 42: What is the behavior of \textstyle \text{∫}_{0}^{1} x^{-1/3} \text{dx}?

Flashcard 43: Evaluate limit: lim⁡a→0+∫a11xdx\textstyle \lim_{a \to 0^+} \int_{a}^{1} \frac{1}{x} \text{dx}lima→0+​∫a1​x1​dx

Flashcard 44: Identify the singularity: ∫011x dx\textstyle \int_{0}^{1} \frac{1}{x} \, dx∫01​x1​dx.

Flashcard 45: What is the behavior of ∫−∞0ex dx\int_{-\infty}^{0} e^x \, dx∫−∞0​exdx?