Integration of x^n e^(−x)dx The question is a definite integral int^inf_0 (x^n)/(e^x)dx So, I'm integrating it by parts, and going by the LAITE principle, I get: Somehow, I'm not exactly sure why I feel like the limit of the first term would go to zero (or n!), given that formula, but I'm not exactly sure how to substitute the n! formula for ex. How do I proceed from here? The options are n!−nIn−1 n!+nIn−1 nIn−1 none of these

Justine Pennington

Justine Pennington

Answered question

2022-11-29

Integration of x n e x d x
The question is a definite integral
0 x n e x d x
So, I'm integrating it by parts, and going by the LAITE principle, I get:
I n = x n e x + 0 n x n 1 e x d x
The value inside the integral is n I n 1 , so all I'm left is to get the limit of the first term as x ranges from 0 to infinity.
x n e x
Somehow, I'm not exactly sure why I feel like the limit of the first term would go to zero (or n!), given that formula, but I'm not exactly sure how to substitute the n! formula for e x . How do I proceed from here?
The options are
n ! n I n 1
n ! + n I n 1
n I n 1
none of these

Answer & Explanation

Alma Garner

Alma Garner

Beginner2022-11-30Added 15 answers

By parts,
x = 0 x n e x d x = x n e x | x = 0 + n x = 0 x n 1 e x d x .
The first term vanishes so that
I n = n I n 1 .
As the base case is
I 0 = x = 0 e x d x = e x | x = 0 = 1 ,
we get the superb formula
I n = n !
Note that as e x contains all powers of x,
e x = k = 0 x k k ! x n + 1 ( n + 1 ) ! ,
and
0 x n e x 1 ( n + 1 ) ! x ,
allowing the limit at to cancel.

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