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

question 14

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

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