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

question 382

Multiple Choice

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

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

---

Definitions:

Bridge

A structure built to span a physical obstacle, such as a body of water, valley, or road, without blocking the way underneath.

Base Level

The lowest point to which erosion by running water can occur, often the level at which a river meets the sea or lake.

Climate

The long-term pattern of weather in a particular area, typically averaged over a period of 30 years.

Stream-Cut Canyon

A deep valley formed through the prolonged action of a stream cutting through the terrain, often displaying steep sides.