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

Question

Quizplus · Accepted Answer

We will use integration by parts with $u = x^n$ and $dv = e^{-ax}dx$:

$$\int x^n e^{-ax}dx = -\frac{x^n}{a}e^{-ax}+\int \frac{nx^{n-1}}{a}e^{-ax}dx$$

Using the same substitution, we get:

$$= -\frac{x^n}{a}e^{-ax}+\frac{n}{a}\int x^{n-1}e^{-ax}dx$$

Therefore, the correct option is (B).

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