Evaluate the Integral $\int f ^ { - 1 } ( x ) d x = x f ^ { - 1 } ( x ) - \int f ( y ) d y , y = f ^ { - 1 } ( x )$

question 9

Multiple Choice

Evaluate the integral.
-Use the formula $\int f ^ { - 1 } ( x ) d x = x f ^ { - 1 } ( x ) - \int f ( y ) d y , y = f ^ { - 1 } ( x )$ to evaluate the integral.
$\int \cot ^ { - 1 } x d x$

• A) \(x \cot ^ { - 1 } x - \ln \sin \left( \cot ^ { - 1 } x \right) + C\)
• B) \(x \cot ^ { - 1 } x - \ln \sin x + C\)
• C) \(x \cot ^ { - 1 } x - x + C\)
• D) \(x \cot ^ { - 1 } x + \ln \sin \left( \cot ^ { - 1 } x \right) + C\)

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Definitions:

Marginal Decision Rule

A principle that states that an action should be taken if, and only if, the marginal benefits exceed the marginal costs.

MC > MR

A condition where a firm's marginal cost is greater than its marginal revenue, suggesting that it would not be profitable to increase output further.

Monopolistic Competition

A market structure where many companies sell products that are similar but not identical, leading to competition based on price, quality, and marketing.

Marginal Decision Rule

A principle stating that actions should be taken if marginal benefits exceed marginal costs.