Simply put, the exponential function is its own derivative by definition. Since indefinite integration is antidifferentation, that means is its own indefinite integral (plus an integration constant).
I have been wondering Why does the integral of e^x is e^x+c. I just been told by the teacher is something to remember but I dont understand it. The graph(y=e^x) shows the line shooting up infinity,how does the area is the same as the line?Is it because it is shooting up really fast so the area of the infinity bit is zero? Thank you.
The integral of e^x is a function whose derivative is e^x because by definition integration is the inverse of differentiation. One such function (whose derivative is e^x) is e^x. All other functions simply differ from this function by a constant (which can be proved using MVTD).
So essentially your asking why the derivative of e^x is e^x. That derivative can be done using the chain rule for differentiation along with the derivative of ln(x) which is 1/x.