Identify U And $d u / d x$ For the Integral $\int \left( 8 + \frac { 1 } { x ^ { 8 } } \right) ^ { 8 } \left( - \frac { 8 } { x ^ { 9 } } \right) d x$

question 6

Multiple Choice

Identify u and $d u / d x$ for the integral $\int \left( 8 + \frac { 1 } { x ^ { 8 } } \right) ^ { 8 } \left( - \frac { 8 } { x ^ { 9 } } \right) d x$ .

• A) \(u = 8 + \frac { 1 } { x ^ { 8 } }\) and \(d u / d x = \frac { 8 } { x ^ { 9 } }\)
• B) \(u = \left( 8 + \frac { 1 } { x ^ { 8 } } \right) ^ { 7 }\) and \(du / d x = - \frac { 8 } { x ^ { 9 } }\)
• C) \(u = 8 + \frac { 1 } { x ^ { 8 } }\) and \(d u/ d x = - \frac { 8 } { x ^ { 9 } }\)
• D) \(u = 8 + \frac { 1 } { x ^ { 8 } }\) and \(d u / d x = - \frac { 9 } { x ^ { 9 } }\)
• E) \(u = \left( 8 + \frac { 1 } { x ^ { 8 } } \right) ^ { 7 }\) and \(d u / d x = \frac { 8 } { x ^ { 9 } }\)

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Definitions:

Marginal Cost

The additional cost incurred by producing one more unit of a good or service; essential for making efficient production and pricing decisions.

Output

The amount of goods or services produced by a business, industry, or country.

Profit-Maximizing Rule

A principle that firms apply to determine the level of output at which they will realize the maximum possible profits.

Short-Run Monopoly

A scenario where a single supplier dominates the market for a brief period, often due to patents, resources, or technology.