Selecting Techniques for Antidifferentiation - AP Calculus AB
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Find the antiderivative of $xe^{x}$.
Find the antiderivative of $xe^{x}$.
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Integration by parts. Product of polynomial and exponential requires integration by parts.
Integration by parts. Product of polynomial and exponential requires integration by parts.
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What is the antiderivative of $\frac{1}{x}$?
What is the antiderivative of $\frac{1}{x}$?
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$\text{ln}|x| + C$. The derivative of $\text{ln}|x|$ is $\frac{1}{x}$.
$\text{ln}|x| + C$. The derivative of $\text{ln}|x|$ is $\frac{1}{x}$.
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What is the antiderivative of $x^{n}$?
What is the antiderivative of $x^{n}$?
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$\frac{x^{n+1}}{n+1} + C$ for $n \neq -1$. Power rule for integration increases exponent by 1 and divides.
$\frac{x^{n+1}}{n+1} + C$ for $n \neq -1$. Power rule for integration increases exponent by 1 and divides.
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Identify the technique for $\frac{1}{\text{sqrt}(x^2 + a^2)}$?
Identify the technique for $\frac{1}{\text{sqrt}(x^2 + a^2)}$?
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Trigonometric substitution. Use $x = a\text{tan}(\theta)$ to simplify the square root.
Trigonometric substitution. Use $x = a\text{tan}(\theta)$ to simplify the square root.
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What is the antiderivative of $\sec(x)\tan(x)$?
What is the antiderivative of $\sec(x)\tan(x)$?
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$\sec(x) + C$. The derivative of $\sec(x)$ is $\sec(x)\tan(x)$.
$\sec(x) + C$. The derivative of $\sec(x)$ is $\sec(x)\tan(x)$.
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What substitution is used for $\text{sqrt}(x^2 + a^2)$?
What substitution is used for $\text{sqrt}(x^2 + a^2)$?
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Trigonometric substitution. Use $x = a\text{tan}(\theta)$ to handle the square root expression.
Trigonometric substitution. Use $x = a\text{tan}(\theta)$ to handle the square root expression.
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Choose the technique for $\text{ln}(a^x)$.
Choose the technique for $\text{ln}(a^x)$.
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Substitution. Simplify $\text{ln}(a^x) = x\text{ln}(a)$ before integrating.
Substitution. Simplify $\text{ln}(a^x) = x\text{ln}(a)$ before integrating.
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Find the antiderivative of $\text{sec}^2(x)$.
Find the antiderivative of $\text{sec}^2(x)$.
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$\text{tan}(x) + C$. The derivative of $\text{tan}(x)$ is $\text{sec}^2(x)$.
$\text{tan}(x) + C$. The derivative of $\text{tan}(x)$ is $\text{sec}^2(x)$.
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What is the antiderivative of $a^x$?
What is the antiderivative of $a^x$?
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$\frac{a^x}{\text{ln}(a)} + C$. General exponential function antiderivative formula with base $a$.
$\frac{a^x}{\text{ln}(a)} + C$. General exponential function antiderivative formula with base $a$.
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Select the technique for $e^{x^2}$.
Select the technique for $e^{x^2}$.
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No elementary antiderivative. Cannot be expressed using elementary functions.
No elementary antiderivative. Cannot be expressed using elementary functions.
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Identify the integration technique for $\text{ln}(x)$.
Identify the integration technique for $\text{ln}(x)$.
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Integration by parts. Use $u = \text{ln}(x)$ and $dv = dx$ for integration by parts.
Integration by parts. Use $u = \text{ln}(x)$ and $dv = dx$ for integration by parts.
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What is the antiderivative of $\text{sin}^2(x)$?
What is the antiderivative of $\text{sin}^2(x)$?
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$\frac{x}{2} - \frac{1}{4}\text{sin}(2x) + C$. Use double angle identity: $\text{sin}^2(x) = \frac{1 - \text{cos}(2x)}{2}$.
$\frac{x}{2} - \frac{1}{4}\text{sin}(2x) + C$. Use double angle identity: $\text{sin}^2(x) = \frac{1 - \text{cos}(2x)}{2}$.
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What is the antiderivative of $\text{cos}^2(x)$?
What is the antiderivative of $\text{cos}^2(x)$?
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$\frac{x}{2} + \frac{1}{4}\text{sin}(2x) + C$. Use double angle identity: $\text{cos}^2(x) = \frac{1 + \text{cos}(2x)}{2}$.
$\frac{x}{2} + \frac{1}{4}\text{sin}(2x) + C$. Use double angle identity: $\text{cos}^2(x) = \frac{1 + \text{cos}(2x)}{2}$.
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Identify the technique for $x^2 \text{ln}(x)$.
Identify the technique for $x^2 \text{ln}(x)$.
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Integration by parts. Polynomial times logarithm requires integration by parts.
Integration by parts. Polynomial times logarithm requires integration by parts.
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Which substitution is used for $\frac{1}{\text{sqrt}(a^2 - x^2)}$?
Which substitution is used for $\frac{1}{\text{sqrt}(a^2 - x^2)}$?
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Trigonometric substitution. Use $x = a\text{sin}(\theta)$ to eliminate the square root.
Trigonometric substitution. Use $x = a\text{sin}(\theta)$ to eliminate the square root.
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What is the antiderivative of $\text{csc}^2(x)$?
What is the antiderivative of $\text{csc}^2(x)$?
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$-\text{cot}(x) + C$. The derivative of $-\text{cot}(x)$ is $\text{csc}^2(x)$.
$-\text{cot}(x) + C$. The derivative of $-\text{cot}(x)$ is $\text{csc}^2(x)$.
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Identify the technique for $\text{arccos}(x)$.
Identify the technique for $\text{arccos}(x)$.
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Integration by parts. Inverse trig functions require integration by parts technique.
Integration by parts. Inverse trig functions require integration by parts technique.
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Which technique is suitable for $\frac{x}{x^2 + 1}$?
Which technique is suitable for $\frac{x}{x^2 + 1}$?
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Substitution. Use $u = x^2 + 1$ to simplify the integrand.
Substitution. Use $u = x^2 + 1$ to simplify the integrand.
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What is the antiderivative of $\text{tan}(x)$?
What is the antiderivative of $\text{tan}(x)$?
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$-\text{ln}|\text{cos}(x)| + C$. Rewrite $\text{tan}(x) = \frac{\text{sin}(x)}{\text{cos}(x)}$ and use substitution.
$-\text{ln}|\text{cos}(x)| + C$. Rewrite $\text{tan}(x) = \frac{\text{sin}(x)}{\text{cos}(x)}$ and use substitution.
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What is the antiderivative of $\frac{1}{x^2}$?
What is the antiderivative of $\frac{1}{x^2}$?
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$-\frac{1}{x} + C$. Rewrite as $x^{-2}$ and apply power rule.
$-\frac{1}{x} + C$. Rewrite as $x^{-2}$ and apply power rule.
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Identify the technique for $e^{3x}\text{sin}(x)$.
Identify the technique for $e^{3x}\text{sin}(x)$.
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Integration by parts. Product of exponential and sine requires integration by parts.
Integration by parts. Product of exponential and sine requires integration by parts.
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What is the antiderivative of $\frac{1}{x^2 + a^2}$?
What is the antiderivative of $\frac{1}{x^2 + a^2}$?
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$\frac{1}{a}\arctan(\frac{x}{a}) + C$. Standard arctangent formula with scaling factor $\frac{1}{a}$.
$\frac{1}{a}\arctan(\frac{x}{a}) + C$. Standard arctangent formula with scaling factor $\frac{1}{a}$.
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Which method is suitable for $x\text{cos}(x)$?
Which method is suitable for $x\text{cos}(x)$?
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Integration by parts. Product of polynomial and cosine requires integration by parts.
Integration by parts. Product of polynomial and cosine requires integration by parts.
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What is the antiderivative of $\frac{1}{\text{sqrt}(1-x^2)}$?
What is the antiderivative of $\frac{1}{\text{sqrt}(1-x^2)}$?
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$\text{arcsin}(x) + C$. Standard inverse sine antiderivative formula.
$\text{arcsin}(x) + C$. Standard inverse sine antiderivative formula.
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Which technique is appropriate for $e^{2x}$?
Which technique is appropriate for $e^{2x}$?
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Substitution. Use $u = 2x$ substitution to handle the coefficient.
Substitution. Use $u = 2x$ substitution to handle the coefficient.
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What is the antiderivative of $e^x$?
What is the antiderivative of $e^x$?
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$e^x + C$. The derivative of $e^x$ is itself, so the antiderivative reverses this.
$e^x + C$. The derivative of $e^x$ is itself, so the antiderivative reverses this.
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Identify the integration technique for $x\text{sin}(x)$.
Identify the integration technique for $x\text{sin}(x)$.
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Integration by parts. Product of $x$ and trig function requires integration by parts.
Integration by parts. Product of $x$ and trig function requires integration by parts.
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State the antiderivative of $\text{sin}(x)$.
State the antiderivative of $\text{sin}(x)$.
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$-\text{cos}(x) + C$. The derivative of $-\text{cos}(x)$ is $\text{sin}(x)$.
$-\text{cos}(x) + C$. The derivative of $-\text{cos}(x)$ is $\text{sin}(x)$.
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What is the antiderivative of $\text{arcsin}(x)$?
What is the antiderivative of $\text{arcsin}(x)$?
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$x \text{arcsin}(x) + \text{sqrt}(1 - x^2) + C$. Standard result from integration by parts formula.
$x \text{arcsin}(x) + \text{sqrt}(1 - x^2) + C$. Standard result from integration by parts formula.
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Choose the integration technique for $\text{arctan}(x)$.
Choose the integration technique for $\text{arctan}(x)$.
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Integration by parts. Inverse trig functions require integration by parts technique.
Integration by parts. Inverse trig functions require integration by parts technique.
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