a)Evaluate the integral ∫ 0 ∞ x n e − x d x for n = 0 , 1 , 2 , 3b)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.

Question

a)Evaluate the integral       ∫    0          ∞            x    n        e          −      x        d  x   for   n  =  0  ,  1  ,  2  ,  3b)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.

Killaninl2 · Accepted Answer

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  !

Dawson Downs · Accepted Answer

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=0Assume that the result (1) is true for n=ki.e.       ∫    0          ∞            x    k        e          −      x        d  x  =  k  !    (2)We have to prove that the result (1) is true for n=k+1i.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!

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.

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New Questions in Calculus 1