a)Evaluate the integral int_0^(oo) x^n e^(-x) dx for n =0, 1, 2, 3. b)Guess the value of int_0^(oo) x^n e^(-x) dx when n is an arbitrary nonnegative integer. c)Prove your guess using mathematical induction.

Libby Owens 2022-07-21 Answered
a)Evaluate the integral 0 x n e x d x  for  n = 0 , 1 , 2 , 3
b)Guess the value of 0 x n e x d x when n is an arbitrary nonnegative integer.
c)Prove your guess using mathematical induction.
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Answers (2)

Killaninl2
Answered 2022-07-22 Author has 20 answers
a)When n=0
0 x n e x d x = 0 e x d x = 1 = 0 !
When n=1
0 x n e x d x = 0 x e x d x = [ x ( e x ) ( 1 ) ( e x ) ] 0 = 1 = 1 !
When n=2
0 x n e x d x = 0 x 2 e x d x = [ x 2 ( e x ) ( 2 x ) ( e x ) + ( 2 ) ( e x ) ] 0 = 2 = 2 !
When n=3
0 x n e x d x = 0 x 3 e x d x = [ x 3 ( e x ) ( 3 x 2 ) ( e x ) + ( 6 x ) ( e x ) ( 6 ) ( e x ) ] 0 = 6 = 3 !
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Dawson Downs
Answered 2022-07-23 Author has 4 answers
b) In general, for non-negative integer,
0 x n e x d x = n ! (1)
c)By part (a), the result (1) is true for n=0
Assume that the result (1) is true for n=k
i.e. 0 x k e x d x = k ! (2)
We have to prove that the result (1) is true for n=k+1
i.e to prove that 0 x k + 1 e x d x = ( k + 1 ) !
Put u = x k + 1 , d v = e x
d u = ( k + 1 ) x k , v = e x
By integration by parts,
0 x k + 1 e x d x = u v 0 v d u = [ ( x k + 1 ) ( e x ) ] 0 + ( k + 1 ) 0 x k e x d x = 0 + ( k + 1 ) ( k ! ) ,  by  ( 2 ) = ( k + 1 ) !
Hence proved!
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