Intergral of the function e^x?

Question

Where x is not necessarily just x. For example. The integral of e^(-50000x). Everything I can think of wants to tell me that the integral of e^(anything) is the same e^(anything). I can't remember what exactly was the rule for integrating e^x when x is something other than x.

Answer 1

You need to use substitution. In the case of e^(-50000x),
let u be -50000x.

So differentiating both sides would give
du/dx=-50000
du/-50000=dx

Int(e^(-50000x)dx) = Int((e^u)du)/-50000 = e^u/-50000 = e^-50000/-50000

Substitution will work for any exponent (even ones as functions of x such as sin(x) or even e^x).

Answer 2

well d/dx e^(-5x) = e^(-5x)*-5

So the antiderivative would be (-1/50000)*e^(-50000x)