If you're seeing this message, it means we're having trouble loading external resources on our website.

If you're behind a web filter, please make sure that the domains *.kastatic.org and *.kasandbox.org are unblocked.

# Indefinite integrals: sums & multiples

AP.CALC:
FUN‑6 (EU)
,
FUN‑6.C (LO)
,
FUN‑6.C.1 (EK)
,
FUN‑6.C.2 (EK)

## Video transcript

so we have listed here are two significant properties of indefinite integrals and we will see in the future that they are very very powerful all this is saying is the indefinite integral of the sum of two different functions is equal to the sum of the indefinite integral of each of those functions this one right over here says the indefinite integral of a constant that's not going to be a function of X of a constant times f of X is the same thing as the constant times the indefinite integral of f of X so one way to think about it is we took the constant out of the integral which we'll see in the future both of these are very useful techniques now if you're satisfied with them as they are written then that's fine you can move on if you want a little bit of a proof what I'm going to do here to give an argument for why this is true is use the derivative properties take the derivative of both sides and see that the Equality holds once we get rid of the integrals so let's do that let's take the derivative with respect to X of both sides of this derivative with respect to X the left side here well this will just become whatever's inside of the indefinite integral this will just become f of X plus G of X plus G of X now what would this become well we could just go to our derivative properties the derivative of the sum of two things that's just the same thing as the sum of the derivatives so this will be a little bit lengthy so this is going to be the derivative with respect to X of this first part plus the derivative with respect to X of this second part and so this first part is the integral of f of X DX we're going to add it and then this is the integral of G of X DX and so let me get write it down this is f of X and then this is G of X now what are these things well these things let me just write this equal sign right over here so in the end this is going to be equal to the derivative of this with respect to X is just going to be f of X and then the derivative with respect to here is just going to be G of X and this is obviously true so now let's tackle this well let's just do the same thing let's take the derivative of both sides so the derivative with respect to X of that and the derivative with respect to X of that so the left hand side will clearly become C times f of X the right hand side is going to become well we know from our derivative properties the derivative of a constant time something is the same thing as the constant times the derivative of that something so then we have the integral indefinite integral of f of X DX and then this thing is just going to be f of X so this is all going to be equal to C times f of X so once again you can see that the quality clearly holds so hopefully this makes you feel good that those properties are true but the more important thing is that you know when to use it so for example if I were to take the integral of let's say x squared plus cosine of X the indefinite integral of that we now know it's going to be useful in the future say well this is the same thing as the integral of x squared DX plus the integral of cosine of X DX so this is the same thing as that plus that and then you can separately evaluate them and this is helpful because we know that if we are trying to figure out the integral of let's say pi times sine of X DX that we can take this constant out pi is in no way dependent on X it's just going to stay being equal to PI so we can take it out and that is going to be equal to PI time's the integral of sine of X two very useful properties and hopefully you feel a lot better about them both now
AP® is a registered trademark of the College Board, which has not reviewed this resource.