Question

how to integrates e^(y/x) in terms of dy?

Answer 1

hint:
you have to consider 1/x as a parameter, for example:
1/x=k
so what is this integral?
\[\Large \int {{e^{ky}}dy} = ...?,\quad k = \frac{1}{x}\]

Answer 2

i dont understand, can you explain more?

Answer 3

for example I can rewrite your function as below:
\[\Large {e^{y/x}} = {e^{y\left( {\frac{1}{x}} \right)}} = {e^{ky}}\]
where:
\[\Large k = \frac{1}{x}\]
now x is not a variable, it is a simple parameter

Answer 4

so you have to compute this integral:
\[\Large \int {{e^{ky}}dy} \]
since:
\[\Large \int {{e^{y/x}}dy} = \int {{e^{ky}}dy} \]

Answer 5

in order to compute the last integral, we have to change variable:
t=ky, where t is the new variable of integration. So our integral becomes:
\[\Large \int {{e^{y/x}}dy} = \int {{e^{ky}}dy} = \int {{e^t}\frac{{dt}}{k}} \]
since we have:
\[\Large t = ky \to dt = kdy \to dy = \frac{{dt}}{k}\]

Answer 6

now:
please, compute this integral:
\[\Large \frac{1}{k}\int {{e^t}dt = ...?} \]

Answer 7

thnks a lot

Answer 8

thanks! :)

Answer 9

just treat x as a constant and do it as you normally would.