# Question Video: Finding the Relation between Two Antiderivatives of the Same Function Mathematics • 12th Grade

If πΉβ (π₯) and πΉβ (π₯) are both antiderivatives of the same function π(π₯), what is the relation between πΉβ and πΉβ?

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### Video Transcript

If πΉ one of π₯ and πΉ two of π₯ are both antiderivatives of the same function π of π₯, what is the relation between πΉ one and πΉ two? (a) πΉ one minus πΉ two is a constant. (b) πΉ one must be the double of πΉ two. (c) πΉ one must be equal to πΉ two. (d) πΉ one plus πΉ 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 πΉ of π₯ plus πΆ such that the first derivative of capital πΉ of π₯, πΉ prime of π₯, is equal to the function π of π₯ and πΆ is any real constant. In other words, an antiderivative of a function lowercase π of π₯ is another function capital πΉ 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.

Weβre given that πΉ one and πΉ two of π₯ are both antiderivatives of the same function πΉ of π₯. So letβs suppose that πΉ one of π₯ is the function πΉ of π₯ plus πΆ one and πΉ two of π₯ is the function πΉ of π₯ plus πΆ two for two constants πΆ one and πΆ 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 πΆ one minus πΆ two, which is a constant. This is option (a) πΉ one minus πΉ two is a constant. Indeed, itβs also true that πΉ two minus πΉ one is a different constant. Letβs consider though the other four options we were offered.

Option (b) states that πΉ one must be the double of πΉ two. This would lead to πΉ of π₯ plus πΆ one is equal to two πΉ of π₯ plus two πΆ two, which may be true for certain values of π₯ in the domain of πΉ, but it wonβt be true for every single value. Option (c) states that πΉ one must be equal to πΉ two. But as weβve seen, antiderivatives are not unique, and there are infinitely many possible antiderivatives for a given function, all differing by a constant.

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