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

Understand the use of pheromones in animal communication and their effects on behavior.

Describe the mechanisms of chemoreception in different species, including humans, and its significance in environmental interaction.

Explain how complex odorant molecules are processed and transmitted to the brain for olfactory perception.

Understand the role of the vomeronasal organ in detecting pheromones and its behavioral effects.

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