### Video Transcript

If πΉ one of π₯ and πΉ two of π₯ are both antiderivatives of the same function π of π₯, what is the relation between πΉ one and πΉ two? (a) πΉ one minus πΉ two is a constant. (b) πΉ one must be the double of πΉ two. (c) πΉ one must be equal to πΉ two. (d) πΉ one plus πΉ two must be a constant. Or (e) only one of them is undefined.

We recall that the general antiderivative of a function lowercase π of π₯ is the function capital πΉ of π₯ plus πΆ such that the first derivative of capital πΉ of π₯, πΉ prime of π₯, is equal to the function π of π₯ and πΆ is any real constant. In other words, an antiderivative of a function lowercase π of π₯ is another function capital πΉ of π₯, whose derivative is equal to the original function. An antiderivative is not unique. And there are many functions which differ up to a constant which give the same derivative.

Weβre given that πΉ one and πΉ two of π₯ are both antiderivatives of the same function πΉ of π₯. So letβs suppose that πΉ one of π₯ is the function πΉ of π₯ plus πΆ one and πΉ two of π₯ is the function πΉ of π₯ plus πΆ two for two constants πΆ one and πΆ two. These functions differ only in the value of their constants. It follows then that if we subtract one function from the other, the result is πΆ one minus πΆ two, which is a constant. This is option (a) πΉ one minus πΉ two is a constant. Indeed, itβs also true that πΉ two minus πΉ one is a different constant. Letβs consider though the other four options we were offered.

Option (b) states that πΉ one must be the double of πΉ two. This would lead to πΉ of π₯ plus πΆ one is equal to two πΉ of π₯ plus two πΆ two, which may be true for certain values of π₯ in the domain of πΉ, but it wonβt be true for every single value. Option (c) states that πΉ one must be equal to πΉ two. But as weβve seen, antiderivatives are not unique, and there are infinitely many possible antiderivatives for a given function, all differing by a constant.

For option (d), πΉ one plus πΉ two would be equal to two πΉ of π₯ plus πΆ one plus πΆ two or two πΉ of π₯ plus a constant πΆ, which is not constant as the value of the function πΉ of π₯ will vary as π₯ varies. This would only be true if πΉ of π₯ is itself a constant function. But it isnβt true in general. For option (e), weβve seen that both functions are defined, so neither is undefined, so we can rule this out too. Our answer is option (a). If πΉ one of π₯ and πΉ two of π₯ are both antiderivatives of the same function lowercase π of π₯, then πΉ one minus πΉ two is a constant.