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# Antiderivatives

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Antiderivatives are the opposite of derivatives. An antiderivative is a function that reverses what the derivative does. One function has many antiderivatives, but they all take the form of a function plus an arbitrary constant. Antiderivatives are a key part of indefinite integrals.

You've been studying derivatives for a long time, that's a huge part of Calculus but a really big part of calculus is to study the antiderivatives so what is an antiderivative? Let's take a look, let's consider the relationship between 2 functions I have written up here, capital F of x=x cubed minus 5x+2 little f of x is 3x squared minus 5 and notice I've given these functions the same letter for the name but this one is capital, this one is lower case so consider them different functions. What's the relationship between these 2, you probably recognize that little f is the derivative of capital f if I differentiate this function I get that one. So that's one thing you'll recognize immediately capital F prime equals little f, so little f is the derivative of capital F so what's the antiderivative relationship?

Well big F would be an antiderivative of little f. So if a function is the derivative of another that first function is an antiderivative of the second. So it's basically the inverse relationship of the derivative relationship but there's one difference between the antiderivative relationship and the derivative relationship and that is there's more than 1 antiderivative. So if you consider this as our function, I just explained that one antiderivative is x cubed minus 5x+2 because this guy's derivative is 3x squared minus 5. But this function's derivative is 3x squared minus 5 as well, so this would be an antiderivative of little f and so will this.

And in fact every function of the form x cubed minus 5x+c has derivative 3x squared minus 5 and so these will all be antiderivatives of this function. So in general there are infinitely many antiderivatives of a given function. If it has an antiderivative it has infinitely many and so we usually represent that fact with a +c, this c means it's a constant it could be any value, any real number value. So any function of this form would be an antiderivative of 3x squared minus 5.

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