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) \(\int x ^ { n } e ^ { x } d x = x ^ { n } e ^ { x } - n \int x ^ { n - 1 } e ^ { x } d x\)

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Definitions:

Coefficient Of Correlation

A measure that indicates the extent to which two variables change together, and the direction of their relationship, with a range from -1 to 1.

Coefficient Of Determination

A statistical measure that assesses the proportion of variance in a dependent variable predictable from an independent variable, often denoted as R².

Coefficient Of Correlation

A measure that determines the degree to which two variables' movements are associated, ranging from -1 to 1.

Independent Variable

A variable that is manipulated in an experiment to observe its effect on the dependent variable, without being influenced by other variables.