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:

Goodness-of-fit Test

A statistical test used to determine how well observed data fit a specific distribution or model.

Test of Independence

A statistical method used to determine if there is a significant association between two categorical variables.

Nonparametric Tests

Statistical tests that do not assume a specific distribution in the data and are used for ordinal or nominal data.

Spearman Rank

A non-parametric measure of rank correlation that assesses how well the relationship between two variables can be described using a monotonic function.