Guessing the Value of an Integral

[Drakkith]

Homework Statement

Guess the value of the following integral when n is an arbitrary positive integer.
Evaluated from 0 to infinity: ∫30xⁿe^-x dx

Homework Equations

The Attempt at a Solution

I've evaluated the integral for values of n from 0 to 3:

n=0: 30
n=1: 30
n=2: 60
n=3: 180

The pattern appears to be that each value is n times larger than the previous value, but I have no idea how to express that mathematically.

 

[Charles Link]

An integration by parts gives F(n)=n*F(n-1) where F(n) is the value of the integral for the value n. Meanwhile, the number 30 is just a constant. The value of F(n) is thereby F(n)=30* (n !) where n !=n(n-1)(n-2)...2*1

 

[Buzz Bloom]

Hi Drakkith:

I think the following will be helpful.

https://en.wikipedia.org/wiki/Factorial​

In particular, take a look at the introduction and also the section

The Gamma and Pi functions.​

Regards,
Buzz

 

[Drakkith]

30n! turned out to work. Thanks!