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

question 123

Multiple Choice

Use integration by parts to establish a reduction formula for the integral.
- $$\int \cos ^ { n } x d x$$

• A) \(\int \cos ^ { n } x d x = \cos ^ { n - 1 } x \sin x - ( n - 1 ) \int \sin x \cos ^ { n - 2 } x d x\)
• B) \(\int \cos ^ { n } x d x = - \cos ^ { n - 1 } x \sin x + ( n - 1 ) \int \cos ^ { n - 2 } x d x\)
• C) \(\int \cos ^ { n } x d x = \frac { 1 } { n } \cos ^ { n - 1 } x \sin x + \frac { n - 1 } { n } \int \cos ^ { n - 2 } x d x\)
• D) \(\int \cos ^ { n } x d x = \frac { 1 } { n } \cos ^ { n - 1 } x \sin x - \frac { n - 1 } { n } \int \cos ^ { n - 1 } x d x\)

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Definitions:

Citric Acid Cycle

is a key metabolic pathway that oxidizes acetyl-CoA to generate ATP and other high-energy molecules, central to cellular energy production.

ATP

Within cells, adenosine triphosphate (ATP) plays a crucial role in energy transfer for metabolism.

Triglycerides Formation

The biochemical process of combining glycerol with three fatty acid molecules, resulting in the primary form of fat storage in animals.

Lipogenesis

The metabolic process by which acetyl-CoA is converted to triglycerides for storage in fat.