Exploring Accumulations of Change - AP Calculus BC
Card 1 of 30
What is the integral of $\frac{1}{\sqrt{1-x^2}}$?
What is the integral of $\frac{1}{\sqrt{1-x^2}}$?
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$\arcsin(x) + C$. Standard inverse trigonometric integration formula.
$\arcsin(x) + C$. Standard inverse trigonometric integration formula.
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What is the integral of $\csc^2(x)$?
What is the integral of $\csc^2(x)$?
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$-\cot(x) + C$. Standard trigonometric integration formula.
$-\cot(x) + C$. Standard trigonometric integration formula.
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What is the integral of $\frac{1}{\cos^2(x)}$?
What is the integral of $\frac{1}{\cos^2(x)}$?
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$\tan(x) + C$. Since $\frac{1}{\cos^2(x)} = \sec^2(x)$.
$\tan(x) + C$. Since $\frac{1}{\cos^2(x)} = \sec^2(x)$.
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What is the antiderivative of $\frac{1}{x}$?
What is the antiderivative of $\frac{1}{x}$?
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$\ln|x| + C$. Natural logarithm is the antiderivative of $\frac{1}{x}$.
$\ln|x| + C$. Natural logarithm is the antiderivative of $\frac{1}{x}$.
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Integrate: $\int (2x + 3) , dx$.
Integrate: $\int (2x + 3) , dx$.
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$x^2 + 3x + C$. Apply power rule to each term separately.
$x^2 + 3x + C$. Apply power rule to each term separately.
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State the integration by parts formula.
State the integration by parts formula.
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$\int u , dv = uv - \int v , du$. Method for integrating products of functions.
$\int u , dv = uv - \int v , du$. Method for integrating products of functions.
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Evaluate: $\int_0^2 (4x^3) , dx$.
Evaluate: $\int_0^2 (4x^3) , dx$.
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$16$. Evaluate $x^4$ from 0 to 2 gives $16 - 0 = 16$.
$16$. Evaluate $x^4$ from 0 to 2 gives $16 - 0 = 16$.
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What is the integral of $\sec(x) \tan(x)$?
What is the integral of $\sec(x) \tan(x)$?
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$\sec(x) + C$. Standard trigonometric integration formula.
$\sec(x) + C$. Standard trigonometric integration formula.
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What is the integral of $\frac{1}{1+x^2}$?
What is the integral of $\frac{1}{1+x^2}$?
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$\arctan(x) + C$. Standard inverse trigonometric integration formula.
$\arctan(x) + C$. Standard inverse trigonometric integration formula.
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Evaluate: $\int_0^3 (x^2) , dx$.
Evaluate: $\int_0^3 (x^2) , dx$.
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$9$. Evaluate $\frac{x^3}{3}$ from 0 to 3 gives $\frac{27}{3} = 9$.
$9$. Evaluate $\frac{x^3}{3}$ from 0 to 3 gives $\frac{27}{3} = 9$.
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Integrate: $\int 5x^4 , dx$.
Integrate: $\int 5x^4 , dx$.
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$x^5 + C$. Apply power rule: $\int 5x^4 dx = 5 \cdot \frac{x^5}{5} = x^5$.
$x^5 + C$. Apply power rule: $\int 5x^4 dx = 5 \cdot \frac{x^5}{5} = x^5$.
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What is the integral of $\tan(x)$?
What is the integral of $\tan(x)$?
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$ -\ln|\cos(x)| + C $. Using substitution $u = \cos(x)$.
$ -\ln|\cos(x)| + C $. Using substitution $u = \cos(x)$.
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What is $C$ in an antiderivative?
What is $C$ in an antiderivative?
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A constant of integration. Represents the family of antiderivatives for indefinite integrals.
A constant of integration. Represents the family of antiderivatives for indefinite integrals.
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What does the definite integral of a positive function represent?
What does the definite integral of a positive function represent?
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The area under the curve $f(x)$ from $x = a$ to $x = b$. Geometric interpretation of definite integrals for positive functions.
The area under the curve $f(x)$ from $x = a$ to $x = b$. Geometric interpretation of definite integrals for positive functions.
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What is the integral of $e^x$?
What is the integral of $e^x$?
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$e^x + C$. The exponential function is its own antiderivative.
$e^x + C$. The exponential function is its own antiderivative.
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State the integral of $\sec^2(x)$.
State the integral of $\sec^2(x)$.
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$\tan(x) + C$. Standard trigonometric integration formula.
$\tan(x) + C$. Standard trigonometric integration formula.
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Calculate $\int_1^2 (2x^2 - 3) , dx$.
Calculate $\int_1^2 (2x^2 - 3) , dx$.
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$\frac{5}{3}$. Evaluate $\frac{2x^3}{3} - 3x$ from 1 to 2.
$\frac{5}{3}$. Evaluate $\frac{2x^3}{3} - 3x$ from 1 to 2.
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Calculate: $\int_0^1 (3x + 2) , dx$.
Calculate: $\int_0^1 (3x + 2) , dx$.
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$3.5$. Evaluate $\frac{3x^2}{2} + 2x$ from 0 to 1 gives $\frac{3}{2} + 2 = 3.5$.
$3.5$. Evaluate $\frac{3x^2}{2} + 2x$ from 0 to 1 gives $\frac{3}{2} + 2 = 3.5$.
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What is the antiderivative of $f(x) = e^x$?
What is the antiderivative of $f(x) = e^x$?
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$F(x) = e^x + C$. The exponential function is its own antiderivative.
$F(x) = e^x + C$. The exponential function is its own antiderivative.
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What is the substitution method in integration?
What is the substitution method in integration?
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Substitute $u = g(x)$ to simplify the integral. Technique for simplifying complex integrals by changing variables.
Substitute $u = g(x)$ to simplify the integral. Technique for simplifying complex integrals by changing variables.
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What is the integral of $\cos(x)$?
What is the integral of $\cos(x)$?
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$\sin(x) + C$. Standard trigonometric integration formula.
$\sin(x) + C$. Standard trigonometric integration formula.
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What is the integral of $\frac{1}{x}$?
What is the integral of $\frac{1}{x}$?
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$\ln|x| + C$. Natural logarithm is the antiderivative of $\frac{1}{x}$.
$\ln|x| + C$. Natural logarithm is the antiderivative of $\frac{1}{x}$.
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State the integral of $\frac{1}{x^2 + 1}$.
State the integral of $\frac{1}{x^2 + 1}$.
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$\arctan(x) + C$. Standard inverse trigonometric integration formula.
$\arctan(x) + C$. Standard inverse trigonometric integration formula.
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State the power rule for integration.
State the power rule for integration.
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$\int x^n , dx = \frac{x^{n+1}}{n+1} + C$, $n \neq -1$. Fundamental integration rule for polynomial functions.
$\int x^n , dx = \frac{x^{n+1}}{n+1} + C$, $n \neq -1$. Fundamental integration rule for polynomial functions.
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What is the integral of $\sin(x)$?
What is the integral of $\sin(x)$?
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$ -\cos(x) + C $. Standard trigonometric integration formula.
$ -\cos(x) + C $. Standard trigonometric integration formula.
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What is the integral of $\csc(x) \cot(x)$?
What is the integral of $\csc(x) \cot(x)$?
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$ -\csc(x) + C $. Standard trigonometric integration formula.
$ -\csc(x) + C $. Standard trigonometric integration formula.
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What is the integral of $\csc^2(x)$?
What is the integral of $\csc^2(x)$?
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$-\cot(x) + C$. Standard trigonometric integration formula.
$-\cot(x) + C$. Standard trigonometric integration formula.
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What is $C$ in an antiderivative?
What is $C$ in an antiderivative?
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A constant of integration. Represents the family of antiderivatives for indefinite integrals.
A constant of integration. Represents the family of antiderivatives for indefinite integrals.
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What does the definite integral of a positive function represent?
What does the definite integral of a positive function represent?
Tap to reveal answer
The area under the curve $f(x)$ from $x = a$ to $x = b$. Geometric interpretation of definite integrals for positive functions.
The area under the curve $f(x)$ from $x = a$ to $x = b$. Geometric interpretation of definite integrals for positive functions.
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What is the integral of $e^x$?
What is the integral of $e^x$?
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$e^x + C$. The exponential function is its own antiderivative.
$e^x + C$. The exponential function is its own antiderivative.
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