How to find the integral of e^(y/x) dy?

Question

anonymous · Answer

namaste

anonymous · Answer

If you're just integrating with respect to y, then 1/x is just a coefficient of y in the exponent of e, meaning we can integrate it just like any other exponential function:

$u = \frac{y}{x}$
$du = \frac{1}{x}dy$
$dy = xdu$

So our integral becomes:

$$x \int\limits_{}^{}e^{u}du = x e^{u} + C = x e^{y/x} + C$$

We're integrating with respect to y, so the derivative of y/x with respect to y is 1/x. Furthermore, since x is a constant, I'm allowed to take it out of the integral and just keep it as a coefficient once I'm done integrating. It's just getting used to the idea of treating variables as constant or ignoring variables you don't need to pay attention to.