currently I'm trying to exercise function integrals and got stuck at a particular function which I found on the internet:
$\int e^{\frac{-s^2}{4k}} ds$
Sadly, there is no solution online. This my attempt so far:
I'm sure that the original function was derived by using the product rule. This would lead to an antiderivate of the form
$z(s)e^{\frac{-s^2}{4k}} + c$
I thought that the function $z$ must depend on $s$ since if the exponential function is derived there will be a $s$ dependent term which can only be cancelled out by having the variable $s$ in it.
Deriving the antiderivate would then lead to (product rule):
$\frac{-s}{2k}z(s)e^{\frac{-s^2}{4k}} + z(s)'e^{\frac{-s^2}{4k}}$
Which can be written as
$e^{\frac{-s^2}{4k}}(\frac{-s}{2k}z(s) + z(s)')$
To get the function in the integral $\frac{-s}{2k}z(s) + z(s)'$ should equal 1. This leads me to the following linear differential equation
$\frac{-s}{2k}z(s) + z(s)' = 1$
Since this is the same as writing $y' + p(x)y = q(x)$ I tried to solve it using an integrating factor:
$\int \frac{-s}{2k} ds \Rightarrow e^\frac{-s^2}{4k}$
I then multiplied both sides with the integrating factor and integrate the result:
$\int (e^\frac{-s^2}{4k}(\frac{-s}{2k}z(s) + z(s)')) ds = \int e^\frac{-s^2}{4k} ds$
$\int (e^\frac{-s^2}{4k}\frac{-s}{2k}z(s) + e^\frac{-s^2}{4k}z(s)') ds = \int e^\frac{-s^2}{4k} ds$
For the left side the product rule implies:
$e^\frac{-s^2}{4k}z(s) = \int e^\frac{-s^2}{4k} ds$
and this is were I got stuck. The right side is exactly my original problem. Integration would then lead to the same result and I will not be able to solve this function.
Can you guys tell me what I'm missing? Is there a special way to treat such integrals? Is the method with the integrating factor the wrong way to go here?
I'm sorry for my lacking knowledge but I'm trying to catch up with mathematics and exercising by myself is hard if there is no further explanation available.
Thanks