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

question 28

Multiple Choice

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

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