Integrating Using Integration By Parts - AP Calculus BC
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Calculate $v$ if $dv = e^x , dx$.
Calculate $v$ if $dv = e^x , dx$.
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$v = e^x$. The antiderivative of $e^x$ is $e^x$.
$v = e^x$. The antiderivative of $e^x$ is $e^x$.
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What is a key benefit of the tabular method?
What is a key benefit of the tabular method?
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Simplifies repeated integration by parts. Reduces calculation time and errors for polynomial-exponential products.
Simplifies repeated integration by parts. Reduces calculation time and errors for polynomial-exponential products.
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What is the integration by parts formula?
What is the integration by parts formula?
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$\int u , dv = uv - \int v , du$. The fundamental formula for integration by parts.
$\int u , dv = uv - \int v , du$. The fundamental formula for integration by parts.
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What is the first step in integration by parts for $\int x \ln(x) , dx$?
What is the first step in integration by parts for $\int x \ln(x) , dx$?
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Choose $u = \ln(x)$ and $dv = x , dx$. Choose $\ln(x)$ as $u$ since it simplifies when differentiated.
Choose $u = \ln(x)$ and $dv = x , dx$. Choose $\ln(x)$ as $u$ since it simplifies when differentiated.
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Identify $u$ for $\int x^2 e^x , dx$ using integration by parts.
Identify $u$ for $\int x^2 e^x , dx$ using integration by parts.
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$u = x^2$. Choose $x^2$ as $u$ since it simplifies when differentiated.
$u = x^2$. Choose $x^2$ as $u$ since it simplifies when differentiated.
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Find $v$ if $dv = \cos(x) , dx$.
Find $v$ if $dv = \cos(x) , dx$.
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$v = \sin(x)$. The antiderivative of $\cos(x)$ is $\sin(x)$.
$v = \sin(x)$. The antiderivative of $\cos(x)$ is $\sin(x)$.
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Which method simplifies repeated parts integration?
Which method simplifies repeated parts integration?
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Tabular integration. Alternative name for the tabular method of integration by parts.
Tabular integration. Alternative name for the tabular method of integration by parts.
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Find $v$ if $dv = e^x , dx$.
Find $v$ if $dv = e^x , dx$.
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$v = e^x$. The antiderivative of $e^x$ is $e^x$.
$v = e^x$. The antiderivative of $e^x$ is $e^x$.
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What rule can simplify repeated integration by parts?
What rule can simplify repeated integration by parts?
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Tabular method. Systematic approach for multiple applications of integration by parts.
Tabular method. Systematic approach for multiple applications of integration by parts.
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Find $v$ if $dv = \sin(x) , dx$.
Find $v$ if $dv = \sin(x) , dx$.
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$v = -\cos(x)$. The antiderivative of $\sin(x)$ is $-\cos(x)$.
$v = -\cos(x)$. The antiderivative of $\sin(x)$ is $-\cos(x)$.
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What is $\frac{d}{dx}(x^2)$ when using integration by parts?
What is $\frac{d}{dx}(x^2)$ when using integration by parts?
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$du = 2x , dx$. The derivative of $x^2$ is $2x$.
$du = 2x , dx$. The derivative of $x^2$ is $2x$.
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What is $\frac{d}{dx}(x)$ when using integration by parts?
What is $\frac{d}{dx}(x)$ when using integration by parts?
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$du = dx$. The derivative of $x$ is $1$.
$du = dx$. The derivative of $x$ is $1$.
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Find the integral of $x^2 e^x$ using integration by parts.
Find the integral of $x^2 e^x$ using integration by parts.
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$x^2 e^x - 2 \int x e^x , dx$. First application of integration by parts, requires second iteration.
$x^2 e^x - 2 \int x e^x , dx$. First application of integration by parts, requires second iteration.
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What is the integral of $x \cos(x)$ using integration by parts?
What is the integral of $x \cos(x)$ using integration by parts?
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$x \sin(x) + \cos(x) + C$. Result after applying integration by parts to $\int x \cos(x) dx$.
$x \sin(x) + \cos(x) + C$. Result after applying integration by parts to $\int x \cos(x) dx$.
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Which part is chosen as $dv$ in integration by parts?
Which part is chosen as $dv$ in integration by parts?
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The part that is easy to integrate. Choose the function that can be integrated easily.
The part that is easy to integrate. Choose the function that can be integrated easily.
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What is the result of $\int \ln(x) , dx$ using integration by parts?
What is the result of $\int \ln(x) , dx$ using integration by parts?
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$x \ln(x) - x + C$. Set $u = \ln(x)$ and $dv = dx$, then apply the formula.
$x \ln(x) - x + C$. Set $u = \ln(x)$ and $dv = dx$, then apply the formula.
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What is the integral of $x^3 e^x$ using integration by parts?
What is the integral of $x^3 e^x$ using integration by parts?
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$x^3 e^x - 3 \int x^2 e^x , dx$. First application of integration by parts, requires further iterations.
$x^3 e^x - 3 \int x^2 e^x , dx$. First application of integration by parts, requires further iterations.
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Identify $u$ for $\int x \sin(x) , dx$ using integration by parts.
Identify $u$ for $\int x \sin(x) , dx$ using integration by parts.
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$u = x$. Choose $x$ as $u$ since it simplifies when differentiated.
$u = x$. Choose $x$ as $u$ since it simplifies when differentiated.
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Determine $dv$ for $\int e^x \sin(x) , dx$ using integration by parts.
Determine $dv$ for $\int e^x \sin(x) , dx$ using integration by parts.
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$dv = e^x , dx$. Choose $e^x dx$ as $dv$ since $e^x$ is easy to integrate.
$dv = e^x , dx$. Choose $e^x dx$ as $dv$ since $e^x$ is easy to integrate.
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Find $\int \arctan(x) , dx$ using integration by parts.
Find $\int \arctan(x) , dx$ using integration by parts.
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$x \arctan(x) - \frac{1}{2} \ln(1 + x^2) + C$. Set $u = \arctan(x)$ and $dv = dx$, then apply the formula.
$x \arctan(x) - \frac{1}{2} \ln(1 + x^2) + C$. Set $u = \arctan(x)$ and $dv = dx$, then apply the formula.
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What is $\frac{d}{dx}(\ln(x))$ when using integration by parts?
What is $\frac{d}{dx}(\ln(x))$ when using integration by parts?
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$du = \frac{1}{x} , dx$. The derivative of $\ln(x)$ is $\frac{1}{x}$.
$du = \frac{1}{x} , dx$. The derivative of $\ln(x)$ is $\frac{1}{x}$.
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Identify $dv$ for $\int x \sin(x) , dx$ using integration by parts.
Identify $dv$ for $\int x \sin(x) , dx$ using integration by parts.
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$dv = \sin(x) , dx$. Choose $\sin(x) dx$ as $dv$ since $\sin(x)$ is easy to integrate.
$dv = \sin(x) , dx$. Choose $\sin(x) dx$ as $dv$ since $\sin(x)$ is easy to integrate.
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Identify $dv$ for $\int x^2 e^x , dx$ using integration by parts.
Identify $dv$ for $\int x^2 e^x , dx$ using integration by parts.
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$dv = e^x , dx$. Choose $e^x dx$ as $dv$ since $e^x$ is easy to integrate.
$dv = e^x , dx$. Choose $e^x dx$ as $dv$ since $e^x$ is easy to integrate.
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Solve $\int x e^{2x} , dx$ using integration by parts.
Solve $\int x e^{2x} , dx$ using integration by parts.
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$\frac{x}{2} e^{2x} - \frac{1}{4} e^{2x} + C$. Apply integration by parts with $u = x$ and $dv = e^{2x} dx$.
$\frac{x}{2} e^{2x} - \frac{1}{4} e^{2x} + C$. Apply integration by parts with $u = x$ and $dv = e^{2x} dx$.
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Determine $u$ for $\int e^x \sin(x) , dx$ using integration by parts.
Determine $u$ for $\int e^x \sin(x) , dx$ using integration by parts.
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$u = \sin(x)$. Choose $\sin(x)$ as $u$ since it cycles when differentiated.
$u = \sin(x)$. Choose $\sin(x)$ as $u$ since it cycles when differentiated.
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Find $v$ if $dv = x , dx$.
Find $v$ if $dv = x , dx$.
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$v = \frac{x^2}{2}$. The antiderivative of $x$ is $\frac{x^2}{2}$.
$v = \frac{x^2}{2}$. The antiderivative of $x$ is $\frac{x^2}{2}$.
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What is the derivative of $\sin(x)$ for integration by parts?
What is the derivative of $\sin(x)$ for integration by parts?
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$du = \cos(x) , dx$. The derivative of $\sin(x)$ is $\cos(x)$.
$du = \cos(x) , dx$. The derivative of $\sin(x)$ is $\cos(x)$.
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Which part is chosen as $u$ in integration by parts?
Which part is chosen as $u$ in integration by parts?
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The part that simplifies when differentiated. Choose the function that becomes simpler when differentiated.
The part that simplifies when differentiated. Choose the function that becomes simpler when differentiated.
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Identify $u$ for $\int x e^x , dx$ using integration by parts.
Identify $u$ for $\int x e^x , dx$ using integration by parts.
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$u = x$. Choose $x$ as $u$ since it simplifies when differentiated.
$u = x$. Choose $x$ as $u$ since it simplifies when differentiated.
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Find the integral of $x^2 e^x$ using the tabular method.
Find the integral of $x^2 e^x$ using the tabular method.
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$x^2 e^x - 2x e^x + 2e^x + C$. Complete solution using repeated integration by parts or tabular method.
$x^2 e^x - 2x e^x + 2e^x + C$. Complete solution using repeated integration by parts or tabular method.
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