Question

Evaluate the following:

If `int_0^oo "e"^(-"a"x^2) x^3  "d"x` = 32, `alpha > 0`, find `alpha`

Answer 1

I = `int_0^oo "e"^(-"a"x^2) x^3  "d"x` 

Put t = x²

dt = 2x dx

`"dt"/2` = x dx

x	0	`oo`
t	0	`oo`

= `int_0^oo "e"^(- omega)"t" "dt"/2 = 1/2 int_0^oo "e"^(- omega) "t"  "dt"`

= `1/2 xx 1/alpha^2`  .........(1)

Given, `int_0^oo "e"^(-alphax^2) x^3  "d"x` = 32

By (1),

`1/2 xx 1/alpha^2` = 32

`1/alpha^2` = 64

`alpha^2 = 1/64`

`alpha = +-  1/8`

⇒ `alpha = 1/8`

∵ `alpha > 0`