Finding specific antiderivatives: exponential function | AP Calculus AB | Khan Academy

We're told that F of 7 is equal to 40 + 5 e 7th power, and f prime of X is equal to 5 e to the X. What is F of 0?

So, to evaluate F of 0, let's take the anti-derivative of f prime of X, and then we're going to have a constant of integration there. So we can use the information that they gave us up here that F of 7 is equal to this. This might look like an expression, but, well, it is an expression; but it's really just a number. There's no variables in this, and so we can use that to solve for our constant of integration. Then we will have fully known what f of X is, and we can use that to evaluate F of 0.

So let's just do it. If f prime of X is equal to 5 e to the X, then F of X is going to be equal to the anti-derivative of f prime of X. So the anti-derivative of 5 e to the X dx.

And this is the thing that I always find amazing about exponentials. Actually, let me just take a step. I'll take that 5 out of the integral so it becomes a little bit more obvious. And so the anti-derivative of e to the X, well, that's just e to the X because the derivative of e to the X is e to the X, which I find amazing every time I have to manipulate or take the derivative or anti-derivative of e to the X.

So this is going to be 5 e to the X + C, and you can verify: take the derivative of 5 e to the X + C. The derivative of 5 e to the X, well, that's 5 e to the X, so that works out, and the derivative—well, and the derivative of C is zero, so you wouldn't see it over here.

So now let's use this information to figure out what C is so that we know exactly what f of X is, and then we can evaluate F of 0. So we know that F of 7, so when X is equal to 7, we're going to—that this expression is going to evaluate to this thing: 40 + 5 e to the 7th power.

So, 5 * e to the 7th power plus C is equal to 40, is equal to 40 + 5 e to the 7th power. And all I did is said, okay, F of 7—well, if this is f of X, let me write this down—if this is F of seven, if this is f of X, I just replace the X with a seven to find F of seven.

We know that F of seven is also going to be equal to that; they gave us that information. But when you just look at this, it's pretty easy to figure out what C is going to be. You can subtract 5 e to the 7th from both sides, and you see that C is equal to 40.

And so we can rewrite F of X. We can say that F of X is equal to 5 e to the X plus C, which is 40. And so now from that, we can evaluate F of 0. F of 0 is going to be 5 * e to the 0 power + 40.

e to the 0 is 1, so it's going to be 5 * 1, which is just 5 + 40, which is equal to 45. And we're done.