Use the Substitution Formula to Evaluate the Integral $$\int _ { 0 } ^ { \pi / 2 } \frac { \cos x } { ( 5 + 4 \sin x ) ^ { 3 } } d x$$

question 127

Multiple Choice

Use the substitution formula to evaluate the integral.
- $$\int _ { 0 } ^ { \pi / 2 } \frac { \cos x } { ( 5 + 4 \sin x ) ^ { 3 } } d x$$

• A) \(- \frac { 3 } { 25 }\)
• B) \(\frac { 28 } { 2025 }\)
• C) \(- \frac { 28 } { 2025 }\)
• D) \(\frac { 7 } { 2025 }\)

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Definitions:

Purely Competitive

A market structure characterized by many small firms producing identical products, where no single firm can influence the market price.

Economic Profit

The difference between total revenue and total costs, including both explicit and implicit costs, representing the profit beyond the normal rate of return.

Average Variable Cost

Average variable cost is the total variable cost divided by the quantity of output produced, representing the variable cost per unit of output.

MR = MC Output

The output level where marginal revenue equals marginal cost, indicating the profit-maximizing level of production for a firm.