Integrating Using Substitution - AP Calculus BC
Card 1 of 30
Identify $du$ for substitution when $u = x^4 + x^2$.
Identify $du$ for substitution when $u = x^4 + x^2$.
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$du = (4x^3 + 2x) , dx$. Differentiate $u = x^4 + x^2$ to get $du$.
$du = (4x^3 + 2x) , dx$. Differentiate $u = x^4 + x^2$ to get $du$.
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What is $du$ if $u = x^2 - 2x + 1$?
What is $du$ if $u = x^2 - 2x + 1$?
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$du = (2x - 2) , dx$. Differentiate $u = x^2 - 2x + 1$ to get $du$.
$du = (2x - 2) , dx$. Differentiate $u = x^2 - 2x + 1$ to get $du$.
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What is $du$ if $u = 3x^2 - 4$?
What is $du$ if $u = 3x^2 - 4$?
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$du = 6x , dx$. Differentiate $u = 3x^2 - 4$ to get $du$.
$du = 6x , dx$. Differentiate $u = 3x^2 - 4$ to get $du$.
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What is $du$ if $u = e^{x^2}$?
What is $du$ if $u = e^{x^2}$?
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$du = 2xe^{x^2} , dx$. Differentiate $u = e^{x^2}$ using chain rule.
$du = 2xe^{x^2} , dx$. Differentiate $u = e^{x^2}$ using chain rule.
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What is $du$ if $u = x^3 + 5x$?
What is $du$ if $u = x^3 + 5x$?
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$du = (3x^2 + 5)dx$. Differentiate $u = x^3 + 5x$ to get $du$.
$du = (3x^2 + 5)dx$. Differentiate $u = x^3 + 5x$ to get $du$.
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Identify $du$ for substitution when $u = e^{x} + x$.
Identify $du$ for substitution when $u = e^{x} + x$.
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$du = (e^{x} + 1) , dx$. Differentiate $u = e^x + x$ to get $du$.
$du = (e^{x} + 1) , dx$. Differentiate $u = e^x + x$ to get $du$.
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What is $du$ if $u = \tan^{-1}(x^2)$?
What is $du$ if $u = \tan^{-1}(x^2)$?
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$du = \frac{2x}{1+x^4} , dx$. Differentiate $u = \tan^{-1}(x^2)$ using chain rule.
$du = \frac{2x}{1+x^4} , dx$. Differentiate $u = \tan^{-1}(x^2)$ using chain rule.
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Identify $du$ for substitution when $u = x^4 + x^2$.
Identify $du$ for substitution when $u = x^4 + x^2$.
Tap to reveal answer
$du = (4x^3 + 2x) , dx$. Differentiate $u = x^4 + x^2$ to get $du$.
$du = (4x^3 + 2x) , dx$. Differentiate $u = x^4 + x^2$ to get $du$.
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Identify $du$ for substitution when $u = \tan^{-1}(e^{x})$.
Identify $du$ for substitution when $u = \tan^{-1}(e^{x})$.
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$du = \frac{e^{x}}{1+e^{2x}} , dx$. Differentiate $u = \tan^{-1}(e^x)$ using chain rule.
$du = \frac{e^{x}}{1+e^{2x}} , dx$. Differentiate $u = \tan^{-1}(e^x)$ using chain rule.
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What is $du$ if $u = x^2 - 2x + 1$?
What is $du$ if $u = x^2 - 2x + 1$?
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$du = (2x - 2) , dx$. Differentiate $u = x^2 - 2x + 1$ to get $du$.
$du = (2x - 2) , dx$. Differentiate $u = x^2 - 2x + 1$ to get $du$.
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What is $du$ if $u = e^{x^2}$?
What is $du$ if $u = e^{x^2}$?
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$du = 2xe^{x^2} , dx$. Differentiate $u = e^{x^2}$ using chain rule.
$du = 2xe^{x^2} , dx$. Differentiate $u = e^{x^2}$ using chain rule.
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Identify $du$ for substitution when $u = e^{x} + x$.
Identify $du$ for substitution when $u = e^{x} + x$.
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$du = (e^{x} + 1) , dx$. Differentiate $u = e^x + x$ to get $du$.
$du = (e^{x} + 1) , dx$. Differentiate $u = e^x + x$ to get $du$.
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What is $du$ if $u = x^3 + 5x$?
What is $du$ if $u = x^3 + 5x$?
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$du = (3x^2 + 5)dx$. Differentiate $u = x^3 + 5x$ to get $du$.
$du = (3x^2 + 5)dx$. Differentiate $u = x^3 + 5x$ to get $du$.
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Identify $du$ for substitution when $u = \frac{1}{x}$.
Identify $du$ for substitution when $u = \frac{1}{x}$.
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$du = -\frac{1}{x^2} dx$. Differentiate $u = \frac{1}{x}$ using power rule.
$du = -\frac{1}{x^2} dx$. Differentiate $u = \frac{1}{x}$ using power rule.
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Identify the most effective choice of $u$ in $\int (5x-1)^7,dx$.
Identify the most effective choice of $u$ in $\int (5x-1)^7,dx$.
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$u=5x-1$. The derivative of $5x-1$ is $5$, a constant factor.
$u=5x-1$. The derivative of $5x-1$ is $5$, a constant factor.
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Evaluate $\int \frac{x}{\sqrt{x^2+4}},dx$ using substitution.
Evaluate $\int \frac{x}{\sqrt{x^2+4}},dx$ using substitution.
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$\sqrt{x^2+4}+C$. Becomes $\frac{1}{2}\int u^{-1/2},du = u^{1/2}+C$.
$\sqrt{x^2+4}+C$. Becomes $\frac{1}{2}\int u^{-1/2},du = u^{1/2}+C$.
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Identify the most effective choice of $u$ in $\int \frac{e^{2x}}{1+e^{2x}},dx$.
Identify the most effective choice of $u$ in $\int \frac{e^{2x}}{1+e^{2x}},dx$.
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$u=1+e^{2x}$. The derivative of $1+e^{2x}$ is $2e^{2x}$, nearly matching the numerator.
$u=1+e^{2x}$. The derivative of $1+e^{2x}$ is $2e^{2x}$, nearly matching the numerator.
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What is the substitution rule for definite integrals, including how to change limits?
What is the substitution rule for definite integrals, including how to change limits?
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$\int_a^b f(g(x))g'(x),dx=\int_{u(a)}^{u(b)} f(u),du$. Change limits: when $x=a$, $u=g(a)$; when $x=b$, $u=g(b)$.
$\int_a^b f(g(x))g'(x),dx=\int_{u(a)}^{u(b)} f(u),du$. Change limits: when $x=a$, $u=g(a)$; when $x=b$, $u=g(b)$.
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Identify the most effective choice of $u$ in $\int \frac{2x}{x^2+5},dx$.
Identify the most effective choice of $u$ in $\int \frac{2x}{x^2+5},dx$.
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$u=x^2+5$. The derivative of $x^2+5$ is $2x$, which appears in the numerator.
$u=x^2+5$. The derivative of $x^2+5$ is $2x$, which appears in the numerator.
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Evaluate $\int \frac{2x}{x^2+5},dx$ using substitution.
Evaluate $\int \frac{2x}{x^2+5},dx$ using substitution.
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$\ln\left|x^2+5\right|+C$. Since $du=2x,dx$, this becomes $\int\frac{du}{u}=\ln|u|+C$.
$\ln\left|x^2+5\right|+C$. Since $du=2x,dx$, this becomes $\int\frac{du}{u}=\ln|u|+C$.
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Identify the most effective choice of $u$ in $\int \cos(3x),dx$.
Identify the most effective choice of $u$ in $\int \cos(3x),dx$.
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$u=3x$. The derivative of $3x$ is $3$, a constant factor we can adjust for.
$u=3x$. The derivative of $3x$ is $3$, a constant factor we can adjust for.
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Evaluate $\int \cos(3x),dx$ using substitution.
Evaluate $\int \cos(3x),dx$ using substitution.
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$\frac{1}{3}\sin(3x)+C$. Since $du=3,dx$, we get $\frac{1}{3}\int\cos(u),du$.
$\frac{1}{3}\sin(3x)+C$. Since $du=3,dx$, we get $\frac{1}{3}\int\cos(u),du$.
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Evaluate $\int (5x-1)^7,dx$ using substitution.
Evaluate $\int (5x-1)^7,dx$ using substitution.
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$\frac{(5x-1)^8}{40}+C$. Using power rule: $\frac{1}{5}\cdot\frac{u^8}{8}$ with $u=5x-1$.
$\frac{(5x-1)^8}{40}+C$. Using power rule: $\frac{1}{5}\cdot\frac{u^8}{8}$ with $u=5x-1$.
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Identify the most effective choice of $u$ in $\int x\sqrt{x^2+9},dx$.
Identify the most effective choice of $u$ in $\int x\sqrt{x^2+9},dx$.
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$u=x^2+9$. The derivative of $x^2+9$ is $2x$, and we have $x$ as a factor.
$u=x^2+9$. The derivative of $x^2+9$ is $2x$, and we have $x$ as a factor.
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Evaluate $\int x\sqrt{x^2+9},dx$ using substitution.
Evaluate $\int x\sqrt{x^2+9},dx$ using substitution.
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$\frac{1}{3}(x^2+9)^{\frac{3}{2}}+C$. Becomes $\frac{1}{2}\int u^{1/2},du$ using power rule.
$\frac{1}{3}(x^2+9)^{\frac{3}{2}}+C$. Becomes $\frac{1}{2}\int u^{1/2},du$ using power rule.
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Identify the most effective choice of $u$ in $\int \frac{x}{\sqrt{x^2+4}},dx$.
Identify the most effective choice of $u$ in $\int \frac{x}{\sqrt{x^2+4}},dx$.
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$u=x^2+4$. The derivative of $x^2+4$ is $2x$, and we have $x$ in the numerator.
$u=x^2+4$. The derivative of $x^2+4$ is $2x$, and we have $x$ in the numerator.
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Evaluate $\int_0^1 2x,e^{x^2},dx$ by substitution with changed limits.
Evaluate $\int_0^1 2x,e^{x^2},dx$ by substitution with changed limits.
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$e-1$. With $u=x^2$, limits change from $[0,1]$ to $[0,1]$; integral is $e^u|_0^1$.
$e-1$. With $u=x^2$, limits change from $[0,1]$ to $[0,1]$; integral is $e^u|_0^1$.
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Evaluate $\int \sec^2(4x),dx$ using substitution.
Evaluate $\int \sec^2(4x),dx$ using substitution.
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$\frac{1}{4}\tan(4x)+C$. Since $\int\sec^2(u),du=\tan(u)+C$ and $du=4,dx$.
$\frac{1}{4}\tan(4x)+C$. Since $\int\sec^2(u),du=\tan(u)+C$ and $du=4,dx$.
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Identify the most effective choice of $u$ in $\int \sec^2(4x),dx$.
Identify the most effective choice of $u$ in $\int \sec^2(4x),dx$.
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$u=4x$. The derivative of $4x$ is $4$, a constant we can factor out.
$u=4x$. The derivative of $4x$ is $4$, a constant we can factor out.
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Evaluate $\int \frac{e^{2x}}{1+e^{2x}},dx$ using substitution.
Evaluate $\int \frac{e^{2x}}{1+e^{2x}},dx$ using substitution.
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$\frac{1}{2}\ln\left|1+e^{2x}\right|+C$. Becomes $\frac{1}{2}\int\frac{du}{u}$ after substitution.
$\frac{1}{2}\ln\left|1+e^{2x}\right|+C$. Becomes $\frac{1}{2}\int\frac{du}{u}$ after substitution.
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