Integrating Using Integration By Parts - AP Calculus BC

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

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

2. What is a key benefit of the tabular method?

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

3. What is the derivative of $\sin(x)$ for integration by parts?

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

4. What is the integral of $x^3 e^x$ using integration by parts?

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

5. Which method simplifies repeated parts integration?

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

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

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

7. Which part is chosen as $dv$ in integration by parts?

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

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

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

9. Find $v$ if $dv = x \, dx$.

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

10. Find $v$ if $dv = e^x \, dx$.

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

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

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

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

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

13. Find $v$ if $dv = \cos(x) \, dx$.

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

14. What is the integration by parts formula?

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

15. What is the result of $\int \ln(x) \, dx$ using integration by parts?

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

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

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

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

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

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

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

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

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

20. What rule can simplify repeated integration by parts?

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

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

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

22. Find $v$ if $dv = \sin(x) \, dx$.

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

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

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

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

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

25. What is $\frac{d}{dx}(\ln(x))$ when using integration by parts?

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

26. Which part is chosen as $u$ in integration by parts?

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

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

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

28. What is the integral of $x \cos(x)$ using integration by parts?

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

29. Find $\int \arctan(x) \, dx$ using integration by parts.

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

30. Find the integral of $x^2 e^x$ using integration by parts.

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

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

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

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