Evaluate the Integral $$\int _ { 0 } ^ { 1 / 2 } 3 \sin ^ { 4 } 2 \pi x d x$$

question 17

Multiple Choice

Evaluate the integral.
- $$\int _ { 0 } ^ { 1 / 2 } 3 \sin ^ { 4 } 2 \pi x d x$$

• A) \(\frac { 9 } { 8 }\)
• B) \(\frac { 9 } { 16 }\)
• C) \(\frac { 9 } { 8 } - \frac { 3 } { 8 \pi }\)
• D) \(\frac { 9 } { 16 } - \frac { 3 } { \pi }\)

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Definitions:

Circadian Rhythms

Physical, mental, and behavioral changes that follow a daily cycle, primarily responding to light and darkness in an organism's environment.

Suprachiasmatic Nucleus

A small region in the brain’s hypothalamus responsible for controlling circadian rhythms, which are the body's internal biological clock.

Pineal Gland

A small endocrine gland in the brain that produces melatonin, a hormone that regulates sleep-wake cycles.

Melatonin

A hormone produced by the pineal gland that regulates sleep and wakefulness cycles.