AP Calculus BC Flashcards: Integrating Using Integration By Parts

What this deck covers

This deck focuses on Integrating Using Integration By Parts, giving you a quick way to review the definitions, rules, and examples that matter most for AP Calculus BC.

How to use these flashcards

Work through these flashcards in short sessions. Try to answer each prompt before flipping the card, then revisit any cards you miss until the explanation feels automatic.

Flashcard 1: What is $\frac{d}{dx}(x)$ when using integration by parts?

Answer: $du = dx$. The derivative of $x$ is $1$.

Flashcard 2: Find the integral of $x^2 e^x$ using the tabular method.

Answer: $x^2 e^x - 2x e^x + 2e^x + C$. Complete solution using repeated integration by parts or tabular method.

Flashcard 3: What is a key benefit of the tabular method?

Answer: Simplifies repeated integration by parts. Reduces calculation time and errors for polynomial-exponential products.

Flashcard 4: Find $v$ if $dv = \sin(x) \, dx$.

Answer: $v = -\cos(x)$. The antiderivative of $\sin(x)$ is $-\cos(x)$.

Flashcard 5: What is the derivative of $\sin(x)$ for integration by parts?

Answer: $du = \cos(x) \, dx$. The derivative of $\sin(x)$ is $\cos(x)$.

Flashcard 6: Determine $dv$ for $\int e^x \sin(x) \, dx$ using integration by parts.

Answer: $dv = e^x \, dx$. Choose $e^x dx$ as $dv$ since $e^x$ is easy to integrate.

Flashcard 7: What is the integral of $x^3 e^x$ using integration by parts?

Answer: $x^3 e^x - 3 \int x^2 e^x \, dx$. First application of integration by parts, requires further iterations.

Flashcard 8: Identify $u$ for $\int x^2 e^x \, dx$ using integration by parts.

Answer: $u = x^2$. Choose $x^2$ as $u$ since it simplifies when differentiated.

Flashcard 9: Which method simplifies repeated parts integration?

Answer: Tabular integration. Alternative name for the tabular method of integration by parts.

Flashcard 10: What is $\frac{d}{dx}(x^2)$ when using integration by parts?

Answer: $du = 2x \, dx$. The derivative of $x^2$ is $2x$.

Flashcard 11: What is the integral of $x^3 e^x$ using integration by parts?

Answer: $x^3 e^x - 3 \int x^2 e^x \, dx$. First application of integration by parts, requires further iterations.

Flashcard 12: Find $v$ if $dv = x \, dx$.

Answer: $v = \frac{x^2}{2}$. The antiderivative of $x$ is $\frac{x^2}{2}$.

Flashcard 13: Identify $u$ for $\int x e^x \, dx$ using integration by parts.

Answer: $u = x$. Choose $x$ as $u$ since it simplifies when differentiated.

Flashcard 14: What is $\frac{d}{dx}(\ln(x))$ when using integration by parts?

Answer: $du = \frac{1}{x} \, dx$. The derivative of $\ln(x)$ is $\frac{1}{x}$.

Flashcard 15: Which part is chosen as $dv$ in integration by parts?

Answer: The part that is easy to integrate. Choose the function that can be integrated easily.

Flashcard 16: What is the integral of $x \cos(x)$ using integration by parts?

Answer: $x \sin(x) + \cos(x) + C$. Result after applying integration by parts to $\int x \cos(x) dx$.

Flashcard 17: Identify $u$ for $\int x^2 e^x \, dx$ using integration by parts.

Answer: $u = x^2$. Choose $x^2$ as $u$ since it simplifies when differentiated.

Flashcard 18: Determine $u$ for $\int e^x \sin(x) \, dx$ using integration by parts.

Answer: $u = \sin(x)$. Choose $\sin(x)$ as $u$ since it cycles when differentiated.

Flashcard 19: Find $v$ if $dv = x \, dx$.

Answer: $v = \frac{x^2}{2}$. The antiderivative of $x$ is $\frac{x^2}{2}$.

Flashcard 20: What is $\frac{d}{dx}(x)$ when using integration by parts?

Answer: $du = dx$. The derivative of $x$ is $1$.

Flashcard 21: Solve $\int x e^{2x} \, dx$ using integration by parts.

Answer: $\frac{x}{2} e^{2x} - \frac{1}{4} e^{2x} + C$. Apply integration by parts with $u = x$ and $dv = e^{2x} dx$.

Flashcard 22: Find $v$ if $dv = e^x \, dx$.

Answer: $v = e^x$. The antiderivative of $e^x$ is $e^x$.

Flashcard 23: Find $v$ if $dv = \cos(x) \, dx$.

Answer: $v = \sin(x)$. The antiderivative of $\cos(x)$ is $\sin(x)$.

Flashcard 24: Determine $u$ for $\int e^x \sin(x) \, dx$ using integration by parts.

Answer: $u = \sin(x)$. Choose $\sin(x)$ as $u$ since it cycles when differentiated.

Flashcard 25: Identify $dv$ for $\int x e^x \, dx$ using integration by parts.

Answer: $dv = e^x \, dx$. Choose $e^x dx$ as $dv$ since $e^x$ is easy to integrate.

Flashcard 26: Find $v$ if $dv = \cos(x) \, dx$.

Answer: $v = \sin(x)$. The antiderivative of $\cos(x)$ is $\sin(x)$.

Flashcard 27: Identify $dv$ for $\int x^2 e^x \, dx$ using integration by parts.

Answer: $dv = e^x \, dx$. Choose $e^x dx$ as $dv$ since $e^x$ is easy to integrate.

Flashcard 28: What is the integration by parts formula?

Answer: $\int u \, dv = uv - \int v \, du$. The fundamental formula for integration by parts.

Flashcard 29: What is the result of $\int \ln(x) \, dx$ using integration by parts?

Answer: $x \ln(x) - x + C$. Set $u = \ln(x)$ and $dv = dx$, then apply the formula.

Flashcard 30: Identify $dv$ for $\int x^2 e^x \, dx$ using integration by parts.

Answer: $dv = e^x \, dx$. Choose $e^x dx$ as $dv$ since $e^x$ is easy to integrate.

Flashcard 31: Identify $u$ for $\int x \sin(x) \, dx$ using integration by parts.

Answer: $u = x$. Choose $x$ as $u$ since it simplifies when differentiated.

Flashcard 32: What is a key benefit of the tabular method?

Answer: Simplifies repeated integration by parts. Reduces calculation time and errors for polynomial-exponential products.

Flashcard 33: Calculate $v$ if $dv = e^x \, dx$.

Answer: $v = e^x$. The antiderivative of $e^x$ is $e^x$.

Flashcard 34: Identify $dv$ for $\int x \sin(x) \, dx$ using integration by parts.

Answer: $dv = \sin(x) \, dx$. Choose $\sin(x) dx$ as $dv$ since $\sin(x)$ is easy to integrate.

Flashcard 35: Identify $dv$ for $\int x \sin(x) \, dx$ using integration by parts.

Answer: $dv = \sin(x) \, dx$. Choose $\sin(x) dx$ as $dv$ since $\sin(x)$ is easy to integrate.

Flashcard 36: What is $\frac{d}{dx}(x)$ when using integration by parts?

Answer: $du = dx$. The derivative of $x$ is $1$.

Flashcard 37: Solve $\int x e^{2x} \, dx$ using integration by parts.

Answer: $\frac{x}{2} e^{2x} - \frac{1}{4} e^{2x} + C$. Apply integration by parts with $u = x$ and $dv = e^{2x} dx$.

Flashcard 38: Find $v$ if $dv = e^x \, dx$.

Answer: $v = e^x$. The antiderivative of $e^x$ is $e^x$.

Flashcard 39: What is $\frac{d}{dx}(x^2)$ when using integration by parts?

Answer: $du = 2x \, dx$. The derivative of $x^2$ is $2x$.

Flashcard 40: What is the first step in integration by parts for $\int x \ln(x) \, dx$?

Answer: Choose $u = \ln(x)$ and $dv = x \, dx$. Choose $\ln(x)$ as $u$ since it simplifies when differentiated.

Flashcard 41: What rule can simplify repeated integration by parts?

Answer: Tabular method. Systematic approach for multiple applications of integration by parts.

Flashcard 42: What is the first step in integration by parts for $\int x \ln(x) \, dx$?

Answer: Choose $u = \ln(x)$ and $dv = x \, dx$. Choose $\ln(x)$ as $u$ since it simplifies when differentiated.

Flashcard 43: What is the derivative of $\sin(x)$ for integration by parts?

Answer: $du = \cos(x) \, dx$. The derivative of $\sin(x)$ is $\cos(x)$.

Flashcard 44: Find $v$ if $dv = \sin(x) \, dx$.

Answer: $v = -\cos(x)$. The antiderivative of $\sin(x)$ is $-\cos(x)$.

Flashcard 45: Which method simplifies repeated parts integration?

Answer: Tabular integration. Alternative name for the tabular method of integration by parts.

Flashcard 46: Identify $dv$ for $\int x e^x \, dx$ using integration by parts.

Answer: $dv = e^x \, dx$. Choose $e^x dx$ as $dv$ since $e^x$ is easy to integrate.

Flashcard 47: What is the result of $\int \ln(x) \, dx$ using integration by parts?

Answer: $x \ln(x) - x + C$. Set $u = \ln(x)$ and $dv = dx$, then apply the formula.

Flashcard 48: Find the integral of $x^2 e^x$ using the tabular method.

Answer: $x^2 e^x - 2x e^x + 2e^x + C$. Complete solution using repeated integration by parts or tabular method.

Flashcard 49: Find the integral of $x^2 e^x$ using integration by parts.

Answer: $x^2 e^x - 2 \int x e^x \, dx$. First application of integration by parts, requires second iteration.

Flashcard 50: Which part is chosen as $u$ in integration by parts?

Answer: The part that simplifies when differentiated. Choose the function that becomes simpler when differentiated.

Flashcard 51: Determine $dv$ for $\int e^x \sin(x) \, dx$ using integration by parts.

Answer: $dv = e^x \, dx$. Choose $e^x dx$ as $dv$ since $e^x$ is easy to integrate.

Flashcard 52: Which part is chosen as $dv$ in integration by parts?

Answer: The part that is easy to integrate. Choose the function that can be integrated easily.

Flashcard 53: What is $\frac{d}{dx}(\ln(x))$ when using integration by parts?

Answer: $du = \frac{1}{x} \, dx$. The derivative of $\ln(x)$ is $\frac{1}{x}$.

Flashcard 54: Calculate $v$ if $dv = e^x \, dx$.

Answer: $v = e^x$. The antiderivative of $e^x$ is $e^x$.

Flashcard 55: Which part is chosen as $u$ in integration by parts?

Answer: The part that simplifies when differentiated. Choose the function that becomes simpler when differentiated.

Flashcard 56: Find $\int \arctan(x) \, dx$ using integration by parts.

Answer: $x \arctan(x) - \frac{1}{2} \ln(1 + x^2) + C$. Set $u = \arctan(x)$ and $dv = dx$, then apply the formula.

Flashcard 57: Identify $u$ for $\int x \sin(x) \, dx$ using integration by parts.

Answer: $u = x$. Choose $x$ as $u$ since it simplifies when differentiated.

Flashcard 58: What is the integration by parts formula?

Answer: $\int u \, dv = uv - \int v \, du$. The fundamental formula for integration by parts.

Flashcard 59: What is the integral of $x \cos(x)$ using integration by parts?

Answer: $x \sin(x) + \cos(x) + C$. Result after applying integration by parts to $\int x \cos(x) dx$.

Flashcard 60: Find $\int \arctan(x) \, dx$ using integration by parts.

Answer: $x \arctan(x) - \frac{1}{2} \ln(1 + x^2) + C$. Set $u = \arctan(x)$ and $dv = dx$, then apply the formula.

Flashcard 61: What rule can simplify repeated integration by parts?

Answer: Tabular method. Systematic approach for multiple applications of integration by parts.

Flashcard 62: Find the integral of $x^2 e^x$ using integration by parts.

Answer: $x^2 e^x - 2 \int x e^x \, dx$. First application of integration by parts, requires second iteration.

Flashcard 63: Identify $u$ for $\int x e^x \, dx$ using integration by parts.

Answer: $u = x$. Choose $x$ as $u$ since it simplifies when differentiated.