How do you evaluate ∫ 0 ∞ x f ( x ) d x, where f ( x ) = x e − x 2 ?

Question

How do you evaluate       ∫    0    ∞    x  f  (  x  )  d  x, where   f  (  x  )  =  x      e          −              x        2             ?

Miriam Payne · Accepted Answer

Use the substitution   x  =      1          2        t to write      ∫    0    ∞        x    2        e          −              x        2              d  x  =      1          2              2                  ∫    0    ∞        t    2        e          −              1        2                    t        2              d  t  ,which by evenness of the integrand is      1          4              2                  ∫          −      ∞        ∞        t    2        e          −              1        2                    t        2              d  t  ,which is equivalent to            π        4        (          1              2        π                    ∫              −        ∞            ∞              t      2              e              −                  1          2                          t          2                      d    t    )    ,and the expression inside parentheses is just the variance of a standard normal random variable, which is 1. Thus, the integral is       π        /    4.

How do you evaluate <msubsup> ∫ 0 <mi mathvariant="normal">∞<

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New Questions in Calculus 1