Question

∫e^(-mx).x^(7)dx Limits zero to infinity ????

Answer 1

`I=(7!)/m^8`

Explanation:

We want to solve

`I=int_0^oo x^7e^(-mx)dx`

Consider a slightly different problem

`I_0=int_0^oo e^(-mx)dx=1/m`

By repeated differentiation of both sides w.r.t. m
(Notice how the negative sign cancels out each time)

`I_1=int_0^oo xe^(-mx)dx=1/m^2`
`I_2=int_0^oo x^2e^(-mx)dx=(1*2)/m^3`
`I_3=int_0^oo x^3e^(-mx)dx=(1*2*3)/m^4`
`I_4=int_0^oo x^4e^(-mx)dx=(1*2*3*4)/m^5`

We may have spotted the pattern, for the general case

`I_n=int_0^oo x^ae^(-mx)dx=(a!)/m^(a+1)`

Or for your case `color(blue)(a=7`

`I_7=int_0^oo x^7e^(-mx)dx=(7!)/m^8`

Bonus info

Consider the integral we found

`int_0^oo x^ae^(-mx)dx=(a!)/m^(a+1)`

For the case `color(blue)(m=1`, this is what we call the gamma function

`Gamma(a+1)=int_0^oo x^ae^(-x)dx=a!`

This is an extended definition of the factorial

From this definition, results which seems odd by our usually definition can be derived
As example

`(0!)=1` and `(1/2!)=sqrt(pi)/2`