## FreeTrader 4 years ago A basic calculus question: The integral of e^2x is [(e^2x)/2] + C. Why do we only multiply by 1/2 without increasing the power?

1. mike0

you increase the power when it is x to the power of some number, when it is some number to the power or x or 2x, the rules are different.

Can you help me understand why the rule is different for e^2x than for x^2?

3. mike0

well, integration is (sort of) the opposite of differentiation, do you know why the derivative of x^2 is 2x?

Yes, I understand the concept.

This is an analysis question more than it is a calculus question, I suppose. Looking for what is happening under the hood conceptually to make the exp function work differently when integrating that the way other functions like the power function for instance.

6. suzi20

e^x will stay e^x when integrated or derivative because e is a number but the power is variable

Suzi20, okay, so if I had x^2N would I not increment that exponent when integrating because it includes the variable N?

8. mike0

if you are integrating with respect to x, then n is not a variable, it is a constant. this is really tough to explain, maybe try khan acadmy, http://www.khanacademy.org/ they have a video that proves that the derivative of e^x is e^x.