๐ Topic Summary
Integration by Parts is a technique used to integrate the product of two functions. When dealing with definite integrals, we use the same formula, but we must evaluate the resulting terms at the limits of integration. The formula is $\int_a^b u \, dv = [u \, v]_a^b - \int_a^b v \, du$, where $u$ and $dv$ are chosen strategically to simplify the integral. Remember to carefully evaluate $[u \, v]$ at both the upper and lower limits ($b$ and $a$).
๐ค Part A: Vocabulary
Match the terms with their definitions:
| Term |
Definition |
| 1. Integration by Parts |
A. The function that remains after differentiating $u$. |
| 2. Definite Integral |
B. The interval over which the definite integral is evaluated. |
| 3. $du$ |
C. A technique to integrate the product of two functions. |
| 4. Limits of Integration |
D. The function integrated after selecting $dv$. |
| 5. $v$ |
E. An integral with upper and lower bounds, resulting in a numerical value. |
โ๏ธ Part B: Fill in the Blanks
Complete the following paragraph with the correct words:
When using Integration by Parts for definite integrals, we choose functions $u$ and $dv$ such that the integral $\int v \, du$ is __________ than the original integral $\int u \, dv$. After applying the formula, $\int_a^b u \, dv = [u \, v]_a^b - \int_a^b v \, du$, we must __________ the term $[u \, v]$ at the __________ and lower limits of integration. This gives us a __________ value for the definite integral.
๐ง Part C: Critical Thinking
Explain in your own words why the choice of $u$ and $dv$ is crucial when applying Integration by Parts. Provide an example to illustrate your point.
