Half integral of x
In my previous post I showed you how to take the half derivative of x. Here, I am going to explain how to find the half integral of x.
We can write the half integral of x as
This is a way I’ve come up with to represent the half integral of a function, as I couldn’t find anything about it.
We can write the second, third, fourth integral of a function as
and, in general,
So we will write the half integral of x like this:
Here we found that
And, in particular, that
This is the half derivative of x, i.e. the inverse of the half integral of x.
We know that the integral of x is x²/2 and that the derivative of x² is 2x. As you can see, the coefficient of the x in the result of the integral is the reciprocal of that of the derivative. This means that the coefficient of the x in result of the half integral of x is
What we need to find now is the exponent of the x. We can give a look at these integrals:
As you can see, the exponent of the x in the result is n (the order of the integral) + 1. This means the coefficient of the x for the half integral is 1/2+1=3/2.
In the previous post we saw that (-0.5)! = sqrt(pi), so
This is the answer!
I really hope you liked this lesson and, if you have any questions, leave a comment and I will be happy to help!
how is (-1/2)! √pi. What does that even mean?
The factorial is defined with the gamma function so that it can be evaluated for any number, be it real or complex!