Use integration by parts to establish a reduction formula for the integral.
-$$\int \mathrm { x } ^ { \mathrm { n } } \mathrm { e } ^ { - a x } \mathrm { dx }$$
A) $\int x ^ { n } e ^ { - a x } d x = \frac { x ^ { n } e ^ { - a x } } { a } - \frac { n } { a } \int x ^ { n - 1 } e ^ { - a x } d x$
B) $\int x ^ { n } e ^ { - a x } d x = - a x ^ { n } e ^ { - a x } + n a \int x ^ { n - 1 } e ^ { - a x } d x$
C) $\int x ^ { n } e ^ { - a x } d x = - \frac { x ^ { n } e ^ { - a x } } { a } + \frac { n } { a } \int x ^ { n - 2 } e ^ { - a x } d x$
D) $\int x ^ { n } e ^ { - a x } d x = - \frac { x ^ { n } e ^ { - a x } } { a } + \frac { n } { a } \int x ^ { n - 1 } e ^ { - a x } d x$

Question

Quizplus · Accepted Answer

The correct reduction formula is obtained by applying integration by parts, where one part is differentiated (x^n) and the other is integrated (e^(-ax)), leading to the formula in option D.

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