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

question 19

Multiple Choice

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

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

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Definitions:

Margin of Safety

The difference between actual or projected sales and the break-even point, indicating the amount by which sales can drop before reaching a loss.

Break-even Point

The point at which total cost and total revenue are equal, resulting in neither profit nor loss.

Break-even Sales

The amount of revenue required to cover total costs, both fixed and variable, at which point a business neither makes a profit nor suffers a loss.

Variable Cost

Overheads that move in sync with the volume of manufacturing or sales, covering labor and material costs.