Question

How do you find the antiderivative of 3^x ?

Answer 1

The antiderivative is basically the same as the indefinite integral.

Do you know the form of the solution to \[\int\limits_{}^{}3^{x}dx\]?

Answer 2

I know how to find the derivative of that, which would be 3^x (ln3). But I'm not sure how to find the antiderivative of it.

Answer 3

Okay.
What exactly is an anti-derivative?
The opposite of a derivative, right?
So, we start by taking the derivative of the function which is f(x) = 3^x
f'(x) = 3^x(ln3), so
You integrate the value to get the antiderivative of the function which would be3^x/(log(3))

Alternatively,
( 3^x) / ln3 + C
--------------------------------------
3^x = e^ {ln (3^x) } = e ^ {x ln3} whose antiderivative is
[e ^ {x ln3}] / ln3 +C
or [3^x] / ln3 +C
or just remember that the derivative of
3^x is (3^x)ln3

Answer 4

Okay but why is it that when we integrate the derivative we divide it by ln3?

Answer 5

I mean I understand from the explanation, but are you supposed to divide by that every time?

Answer 6

I suppose that's the rule. You do the opposite of whatever you would to find its derivative.

Answer 7

Okay thanks! :)