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

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

Flashcard 2: 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 3: What is the integration by parts formula?

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

Flashcard 4: 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 5: 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 6: Find $v$ if $dv = \cos(x) \, dx$.

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

Flashcard 7: Which method simplifies repeated parts integration?

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

Flashcard 8: Find $v$ if $dv = e^x \, dx$.

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

Flashcard 9: What rule can simplify repeated integration by parts?

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

Flashcard 10: Find $v$ if $dv = \sin(x) \, dx$.

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

Flashcard 11: What is $\frac{d}{dx}(x^2)$ when using integration by parts?

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

Flashcard 12: What is $\frac{d}{dx}(x)$ when using integration by parts?

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

Flashcard 13: 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 14: 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 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 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 17: 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 18: 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 19: 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 20: 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 21: 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 22: 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 23: 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 24: 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 25: 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 26: Find $v$ if $dv = x \, dx$.

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

Flashcard 27: 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 28: 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 29: 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 30: 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.