Use Integration by Parts to Establish a Reduction Formula for the Integral

question 304

Multiple Choice

Use integration by parts to establish a reduction formula for the integral.
- $$\int x ^ { n } e ^ { - x ^ { 2 } } d x$$

• A) \(\int x ^ { n } e ^ { - x ^ { 2 } } d x = - 2 x ^ { n - 1 } e ^ { - x ^ { 2 } } - 2 ( n - 1 ) \int x ^ { n - 2 } e ^ { - x ^ { 2 } } d x\)
• B) \(\int \mathrm { x } ^ { \mathrm { n } } \mathrm { e } ^ { - \mathrm { x } ^ { 2 } } \mathrm { dx } = - \frac { 1 } { 2 } \mathrm { x } ^ { \mathrm { n } } \mathrm { e } ^ { - \mathrm { x } ^ { 2 } + \frac { \mathrm { n } } { 2 } \int \mathrm { x } ^ { \mathrm { n } - 1 } \mathrm { e } ^ { - \mathrm { x } ^ { 2 } } \mathrm { dx } }\)
• C) \(\int x ^ { n } e ^ { - x ^ { 2 } } d x = - \frac { 1 } { 2 } x ^ { n - 1 } e ^ { - x ^ { 2 } } + \frac { n - 1 } { 2 } \int x ^ { n - 2 } e ^ { - x ^ { 2 } } d x\)
• D) \(\int \mathrm { x } ^ { \mathrm { n } } \mathrm { e } ^ { - \mathrm { x } ^ { 2 } } \mathrm { dx } = \mathrm { n } \mathrm { x } ^ { \mathrm { n } - 1 } \mathrm { e } ^ { - \mathrm { x } ^ { 2 } + 2 n } \int \mathrm { x } ^ { \mathrm { n } - 1 } \mathrm { e } ^ { - \mathrm { x } ^ { 2 } } \mathrm { dx }\)

---

Definitions:

Analgesia Administration

The process of providing medication or treatment to relieve pain, which may be achieved through various routes and modalities.

Health Disparities

Differences in health status or access to healthcare across different populations, often influenced by social, economic, or environmental factors.

Health Disparity

The differences in health outcomes and their determinants between different segments of the population, often influenced by social, economic, and environmental factors.

Frequent Fast Food

Regular consumption of food from fast-food restaurants, often associated with negative health effects.