Antiderivatives and indefinite integrals

we know how to take derivatives of functions if I apply the derivative operator to x squared I get 2x now if I also apply the derivative operator to x squared plus 1 I also get 2x if I apply the derivative operator to x squared plus PI I also get 2x the derivative of x squared is 2x derivative with respect to X of Pi of a constant is just 0 the derivative with respect to X of 1 is just a constant it's just 0 so once again this is just going to be equal to 2x in general the derivative with respect to X of x squared plus any constant any constant is going to be equal to 2x the derivative of x squared with respect to X is 2x derivative of a constant with respect to X a constant does not change with respect to X so it's just equal to 0 so you have you take the derivative you apply the derivative operator to any of these expressions and you get 2x now let's go the other way around let's think about the antiderivative anti derivative and one way to think about it is the it's the we're doing the opposite of the derivative operator the derivative operator you get an expression and you find its derivative now what we want to do is given some expression we want to find what it could be the derivative of what it could be the derivative of so if someone were to tell you or give you 2x if someone were to say 2x or let me find it write this so if someone were to ask you what is what is 2x the derivative derivative of they're essentially asking you for the anti derivative and so you could say well 2x is the derivative of x squared you could say 2x is the derivative of x squared but you could also say 2x is the derivative of x squared plus 1 you could also say that 2x is the derivative of x squared plus pi I think you get the general idea so if you wanted to write it in the most general sense you would write that 2x is the derivative of x squared plus some constant so this is what you would consider the antiderivative of 2x now that's all nice but this is kind of clumsy to have to write a sentence like this so we let's come up with some notation for the antiderivative and the convention here is to use kind of a strange-looking notation is to use a a big elongated s looking thing like that and the DX around the function that you're trying to take the antiderivative of so in this case it would look something like this this is just saying this is equal to the antiderivative of 2x and the antiderivative of 2x we have already seen is x squared is x squared plus C now you might be saying why do we use this type of crazy notation it'll become more obvious when we study the definite integral and areas under curves and taking sums of rectangles and in order to approximate the area of the curve here it really should just be viewed as a notation for antiderivative and this notation right over here this whole expression is called the indefinite integral the in definite in definite in definite integral of 2x which is another way of just saying the antiderivative of 2x