Integration of an exponential function

[Safinaz]

Homework Statement

Hello ,

How to integrate

Relevant Equations

$\int ~ dy ~ e^{-2 \alpha(y)}$

My trial :

I think $\int ~ dy ~ e^{-2 \alpha(y)}$ dose not simply equal: $- \frac{1}{2}e^{-2 \alpha(y)}$ cause $\alpha$ is a function in $y$.

So any help about the right answer is appreciated!

 

[anuttarasammyak]

What is $\alpha(y)$ ?

 

[Safinaz]

What is $\alpha(y)$ ?

Should I assume it to make the integration, right?
Well, in this case let's assume it increases with y exponentially or it's a slowly varying function

 

[Steve4Physics]

Should I assume it to make the integration, right?
Well, in this case let's assume it increases with y exponentially or it's a slowly varying function

You need to specify the function $\alpha(y)$ - i.e. give the actual formula for $\alpha$ in terms of y. Or (if integrating numerically) you need a table giving values of $\alpha$ for values of y over the range of interest.

If $\alpha(y)$ can be represented as a linear function of y ($\alpha(y) = ay + b$ with a and b as constants) then the integration is clearly simple.

For more complicated functions, I believe there are no general analytical methods, though special cases may have solutions . E.g. with $\alpha(y) = y^2$ the integral can be expressed in terms of the error function – see https://www.wolframalpha.com/input/?i=e^(-2y^2)

Also, see discussion here: https://math.stackexchange.com/questions/19390/integrating-efx

 

[Safinaz]

@Steve4Physics. Hay! just saying thank you very much! The answer is so helpful