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) \(\int \mathrm { x } ^ { \mathrm { n } } \mathrm { e } ^ { - \mathrm { x } ^ { 2 } } \mathrm { dx } = \mathrm { n } \mathrm { x } ^ { \mathrm { n } - 1 } \mathrm { e } ^ { - \mathrm { x } ^ { 2 } + 2 n } \int \mathrm { x } ^ { \mathrm { n } - 1 } \mathrm { e } ^ { - \mathrm { x } ^ { 2 } } \mathrm { dx }\)

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