Antidifferentiation - Concept
Finding the antiderivatives of a function require a little backwards thinking. Since the derivative of the wanted antiderivative is the given function, checking for correctness is easy. You just take the derivative, and see if it is the given function. Also, antiderivatives of functions happen to be not just one function, but a whole family of functions. This family can be written as a polynomial plus c, where c stands for any constant.
I want to talk about how to find antiderivatives so let's recall what an antiderivative is capital F of x is an antiderivative of little f of x means that capital F prime equals little f. So when you look at this relationship little f is the derivative of big f, big F is an antiderivative of little f, so you basically guess and check to find antiderivative that's one method for doing it. So finding antiderivatives of this function for example, f of x, little f of x equals 10x+4 you'd have to think of a function who's derivative is 10+4. Now I can think of x squared has a derivative of 2x, so what I have to do x squared to make the derivative 10x multiply it by 5 so I could start with a capital F which is 5x squared plus and what function is derivative as 4? Well 4x has a derivative of 4, so capital F of x equals 5x squared plus 4x.
Now when you check that by differentiating capital F prime, this would be the 2 would come down in front you get 10x, the derivative of 4x is 4 and there we go that works so this would be an antiderivative of 10x+4. But it's not the only one, 5x squared plus 4x-60, the derivative of the minus 60 part is just 0 so the derivative will still be 10x+4 and another one 5x squared plus 4x+100 any constant you want to add the derivative is going to be 0 so the derivative will still be 10x+4 and you can see I can go on and on like this forever. I can keep coming up with functions and just putting in different values for this constant and I'll have lots and lots of antiderivatives of 10x plus 4.
But that's not generally the way we write our answer, we write our answer in a nice condensed form. Any function of the form 5x squared plus 4x+c is an antiderivative of the function 10x+4. So this is the way we generally write our answer and the important thing to recognize here is once I came up with that first answer 5x squared plus 4x any constant I wanted to add that would still give me an antiderivative. And that leads me to this important theorem, if capital F of x is an antiderivative of little f of x then capital F of x plus c are all the antiderivatives of little f of x. That means that any function of this form is going to be an antiderivative and that's all there are. So that's really powerful and so as I just said the first example I came up with 5x squared plus 4x once I had found that antiderivative of 10x+4 I was done all I have to do is add a +c and that's all the antiderivatives of 10x+4 very powerful theorem once you find one antiderivative add a +c and you've found all of them.