Integrating Using Substitution - AP Calculus AB
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What substitution simplifies the integral of $(2x + 1)^5$?
What substitution simplifies the integral of $(2x + 1)^5$?
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$u = 2x + 1$. Let $u = 2x + 1$ to transform $(2x + 1)^5$ into $u^5$ for easier integration.
$u = 2x + 1$. Let $u = 2x + 1$ to transform $(2x + 1)^5$ into $u^5$ for easier integration.
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What is the substitution for $u$ if $du = -\text{sin } x , dx$?
What is the substitution for $u$ if $du = -\text{sin } x , dx$?
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$u = \text{cos } x$. Since $\frac{d}{dx}(\cos x) = -\sin x$, if $du = -\sin x dx$, then $u = \cos x$.
$u = \text{cos } x$. Since $\frac{d}{dx}(\cos x) = -\sin x$, if $du = -\sin x dx$, then $u = \cos x$.
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What is the integral of $\frac{1}{u} , du$?
What is the integral of $\frac{1}{u} , du$?
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$\text{ln } |u| + C$. The antiderivative of $\frac{1}{u}$ is $\ln|u|$ plus the constant of integration.
$\text{ln } |u| + C$. The antiderivative of $\frac{1}{u}$ is $\ln|u|$ plus the constant of integration.
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Determine $du$ for $u = \text{sin } x$ in substitution.
Determine $du$ for $u = \text{sin } x$ in substitution.
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$du = \text{cos } x , dx$. The derivative of $u = \sin x$ is $\cos x$, so $du = \cos x dx$.
$du = \text{cos } x , dx$. The derivative of $u = \sin x$ is $\cos x$, so $du = \cos x dx$.
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Determine $du$ if $u = \text{tan } x$ in substitution.
Determine $du$ if $u = \text{tan } x$ in substitution.
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$du = \text{sec}^2 x , dx$. The derivative of $u = \tan x$ is $\sec^2 x$, so $du = \sec^2 x dx$.
$du = \text{sec}^2 x , dx$. The derivative of $u = \tan x$ is $\sec^2 x$, so $du = \sec^2 x dx$.
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What is the integral of $u^n , du$?
What is the integral of $u^n , du$?
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$\frac{u^{n+1}}{n+1} + C$, $n \neq -1$. Use the power rule for integration: increase exponent by 1 and divide.
$\frac{u^{n+1}}{n+1} + C$, $n \neq -1$. Use the power rule for integration: increase exponent by 1 and divide.
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What is the integral of $\text{sin } u , du$?
What is the integral of $\text{sin } u , du$?
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$-\text{cos } u + C$. The antiderivative of $\sin u$ is $-\cos u$ plus the constant of integration.
$-\text{cos } u + C$. The antiderivative of $\sin u$ is $-\cos u$ plus the constant of integration.
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What substitution simplifies the integral of $x \text{cos}(x^2)$?
What substitution simplifies the integral of $x \text{cos}(x^2)$?
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$u = x^2$. Let $u = x^2$ so that $du = 2x dx$ matches the $x$ factor in $x\cos(x^2)$.
$u = x^2$. Let $u = x^2$ so that $du = 2x dx$ matches the $x$ factor in $x\cos(x^2)$.
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Determine $du$ if $u = x^2 - 1$ in substitution.
Determine $du$ if $u = x^2 - 1$ in substitution.
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$du = 2x , dx$. The derivative of $u = x^2 - 1$ is $2x$, so $du = 2x dx$.
$du = 2x , dx$. The derivative of $u = x^2 - 1$ is $2x$, so $du = 2x dx$.
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Find $u$ for substitution if $du = \frac{1}{x^2} , dx$.
Find $u$ for substitution if $du = \frac{1}{x^2} , dx$.
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$u = -\frac{1}{x}$. Since $\frac{d}{dx}(-\frac{1}{x}) = \frac{1}{x^2}$, if $du = \frac{1}{x^2} dx$, then $u = -\frac{1}{x}$.
$u = -\frac{1}{x}$. Since $\frac{d}{dx}(-\frac{1}{x}) = \frac{1}{x^2}$, if $du = \frac{1}{x^2} dx$, then $u = -\frac{1}{x}$.
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Identify $u$ for substitution if $du = 2x , dx$.
Identify $u$ for substitution if $du = 2x , dx$.
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$u = x^2$. Since $\frac{d}{dx}(x^2) = 2x$, if $du = 2x dx$, then $u = x^2$.
$u = x^2$. Since $\frac{d}{dx}(x^2) = 2x$, if $du = 2x dx$, then $u = x^2$.
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Find $u$ for substitution if $du = \text{cos } x , dx$.
Find $u$ for substitution if $du = \text{cos } x , dx$.
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$u = \text{sin } x$. Since $\frac{d}{dx}(\sin x) = \cos x$, if $du = \cos x dx$, then $u = \sin x$.
$u = \text{sin } x$. Since $\frac{d}{dx}(\sin x) = \cos x$, if $du = \cos x dx$, then $u = \sin x$.
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What is the integral of $\text{cos } u , du$?
What is the integral of $\text{cos } u , du$?
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$\text{sin } u + C$. The antiderivative of $\cos u$ is $\sin u$ plus the constant of integration.
$\text{sin } u + C$. The antiderivative of $\cos u$ is $\sin u$ plus the constant of integration.
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Find $u$ for substitution if $du = 7x^6 , dx$.
Find $u$ for substitution if $du = 7x^6 , dx$.
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$u = x^7$. Since $\frac{d}{dx}(x^7) = 7x^6$, if $du = 7x^6 dx$, then $u = x^7$.
$u = x^7$. Since $\frac{d}{dx}(x^7) = 7x^6$, if $du = 7x^6 dx$, then $u = x^7$.
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Determine $du$ if $u = \text{ln } x$ in substitution.
Determine $du$ if $u = \text{ln } x$ in substitution.
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$du = \frac{1}{x} , dx$. The derivative of $u = \ln x$ is $\frac{1}{x}$, so $du = \frac{1}{x} dx$.
$du = \frac{1}{x} , dx$. The derivative of $u = \ln x$ is $\frac{1}{x}$, so $du = \frac{1}{x} dx$.
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Determine $du$ if $u = 3x^4 + 2x$ in substitution.
Determine $du$ if $u = 3x^4 + 2x$ in substitution.
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$du = (12x^3 + 2) , dx$. The derivative of $u = 3x^4 + 2x$ is $12x^3 + 2$, so $du = (12x^3 + 2) dx$.
$du = (12x^3 + 2) , dx$. The derivative of $u = 3x^4 + 2x$ is $12x^3 + 2$, so $du = (12x^3 + 2) dx$.
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What substitution simplifies the integral of $x e^{x^2}$?
What substitution simplifies the integral of $x e^{x^2}$?
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$u = x^2$. Let $u = x^2$ so that $du = 2x dx$ matches the $x$ factor in $xe^{x^2}$.
$u = x^2$. Let $u = x^2$ so that $du = 2x dx$ matches the $x$ factor in $xe^{x^2}$.
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Identify $u$ for substitution if $du = \text{sec}^2 x , dx$.
Identify $u$ for substitution if $du = \text{sec}^2 x , dx$.
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$u = \text{tan } x$. Since $\frac{d}{dx}(\tan x) = \sec^2 x$, if $du = \sec^2 x dx$, then $u = \tan x$.
$u = \text{tan } x$. Since $\frac{d}{dx}(\tan x) = \sec^2 x$, if $du = \sec^2 x dx$, then $u = \tan x$.
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Identify $u$ for substitution if $du = e^x , dx$.
Identify $u$ for substitution if $du = e^x , dx$.
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$u = e^x$. Since $\frac{d}{dx}(e^x) = e^x$, if $du = e^x dx$, then $u = e^x$.
$u = e^x$. Since $\frac{d}{dx}(e^x) = e^x$, if $du = e^x dx$, then $u = e^x$.
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Determine $du$ if $u = 2x^3 + 1$ in substitution.
Determine $du$ if $u = 2x^3 + 1$ in substitution.
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$du = 6x^2 , dx$. The derivative of $u = 2x^3 + 1$ is $6x^2$, so $du = 6x^2 dx$.
$du = 6x^2 , dx$. The derivative of $u = 2x^3 + 1$ is $6x^2$, so $du = 6x^2 dx$.
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What is the integral of $e^u , du$?
What is the integral of $e^u , du$?
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$e^u + C$. The antiderivative of $e^u$ is $e^u$ plus the constant of integration.
$e^u + C$. The antiderivative of $e^u$ is $e^u$ plus the constant of integration.
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Identify $du$ if $u = x^3$ in substitution.
Identify $du$ if $u = x^3$ in substitution.
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$du = 3x^2 , dx$. The derivative of $u = x^3$ is $\frac{du}{dx} = 3x^2$, so $du = 3x^2 dx$.
$du = 3x^2 , dx$. The derivative of $u = x^3$ is $\frac{du}{dx} = 3x^2$, so $du = 3x^2 dx$.
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What substitution simplifies the integral of $\text{cos}(x^2)$?
What substitution simplifies the integral of $\text{cos}(x^2)$?
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$u = x^2$. Let $u = x^2$ to simplify the argument of the cosine function.
$u = x^2$. Let $u = x^2$ to simplify the argument of the cosine function.
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Find $u$ for substitution if $du = \frac{1}{x} , dx$.
Find $u$ for substitution if $du = \frac{1}{x} , dx$.
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$u = \text{ln } x$. Since $\frac{d}{dx}(\ln x) = \frac{1}{x}$, if $du = \frac{1}{x} dx$, then $u = \ln x$.
$u = \text{ln } x$. Since $\frac{d}{dx}(\ln x) = \frac{1}{x}$, if $du = \frac{1}{x} dx$, then $u = \ln x$.
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What substitution simplifies the integral of $(2x + 1)^5$?
What substitution simplifies the integral of $(2x + 1)^5$?
Tap to reveal answer
$u = 2x + 1$. Let $u = 2x + 1$ to transform $(2x + 1)^5$ into $u^5$ for easier integration.
$u = 2x + 1$. Let $u = 2x + 1$ to transform $(2x + 1)^5$ into $u^5$ for easier integration.
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What is the substitution for $u$ if $du = -\text{sin } x , dx$?
What is the substitution for $u$ if $du = -\text{sin } x , dx$?
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$u = \text{cos } x$. Since $\frac{d}{dx}(\cos x) = -\sin x$, if $du = -\sin x dx$, then $u = \cos x$.
$u = \text{cos } x$. Since $\frac{d}{dx}(\cos x) = -\sin x$, if $du = -\sin x dx$, then $u = \cos x$.
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Identify $u$ for substitution if $du = 5x^4 , dx$.
Identify $u$ for substitution if $du = 5x^4 , dx$.
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$u = x^5$. Since $\frac{d}{dx}(x^5) = 5x^4$, if $du = 5x^4 dx$, then $u = x^5$.
$u = x^5$. Since $\frac{d}{dx}(x^5) = 5x^4$, if $du = 5x^4 dx$, then $u = x^5$.
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Determine $du$ if $u = \text{ln}(4x)$ in substitution.
Determine $du$ if $u = \text{ln}(4x)$ in substitution.
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$du = \frac{1}{x} , dx$. The derivative of $u = \ln(4x)$ is $\frac{1}{x}$, so $du = \frac{1}{x} dx$.
$du = \frac{1}{x} , dx$. The derivative of $u = \ln(4x)$ is $\frac{1}{x}$, so $du = \frac{1}{x} dx$.
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What is the integral of $u^n , du$?
What is the integral of $u^n , du$?
Tap to reveal answer
$\frac{u^{n+1}}{n+1} + C$, $n \neq -1$. Use the power rule for integration: increase exponent by 1 and divide.
$\frac{u^{n+1}}{n+1} + C$, $n \neq -1$. Use the power rule for integration: increase exponent by 1 and divide.
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Determine $du$ for $u = \text{sin } x$ in substitution.
Determine $du$ for $u = \text{sin } x$ in substitution.
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$du = \text{cos } x , dx$. The derivative of $u = \sin x$ is $\cos x$, so $du = \cos x dx$.
$du = \text{cos } x , dx$. The derivative of $u = \sin x$ is $\cos x$, so $du = \cos x dx$.
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