Anti-derivative vs. indefinite integral?

mikeystipey

hi folks!

I'm studying an integration module and one of the questions on a past paper goes as follows:

Let f be a continuous function. Explain the terms

(i) anti-derivative of f

(ii) indefinite integral of f

I'm tempted to give the answer 'well, they're kind of the same thing', but I suspect that would be rather ill received and more importantly probably wrong. The prescribed text book for our course (Thomas Calculus) gives the following definition for an anti-derivative:

"A function F is an antiderivative of f on an interval I if F'(x)=f(x) for all x in I."

Do you think this would be a sufficient answer, and does this definition imply that the term anti-derivative is used for definite integrals (because they are over a fixed interval)?

The answer I'm thinkinig of for part (ii) is:

An indefinite integral of f produces another function F, as opposed to a definite integral which gives the area under a function f.

Any help would be much appreciated!

TheBody

Go with the definition from the book of an anti derivative. For example, X^2 is an antiderivative of 2x but so is x^2+1 or x^2+2 or x^2+3 etc. In other words, you can differentiate any of these functions and get 2x.

The indefinite integral is the set of all antiderivatives. So using the example above, the integral of 2x is x^2+c where c is the constant of integration. So if you let c=1 or c=2 or c=3 etc you would get the antiderivatives above. (Note that c can take on any value. Not just positive or a whole number)

Hope that makes sense!!!

mikeystipey

TheBody wrote: »

Go with the definition from the book of an anti derivative. For example, X^2 is an antiderivative of 2x but so is x^2+1 or x^2+2 or x^2+3 etc. In other words, you can differentiate any of these functions and get 2x.

The indefinite integral is the set of all antiderivatives. So using the example above, the integral of 2x is x^2+c where c is the constant of integration. So if you let c=1 or c=2 or c=3 etc you would get the antiderivatives above. (Note that c can take on any value. Not just positive or a whole number)

Hope that makes sense!!!

Hi theBody, many thanks for your suggestion here, only saw your reply now. That makes sense all right, particularly where you say 'The indefinite integral is the set of all antiderivatives'. I feel its a rather in depth question for the module I'm studying but what can you do!
cheers
Mikey

TheBody

Your welcome!!

LeixlipRed

mikeystipey wrote: »

Hi theBody, many thanks for your suggestion here, only saw your reply now. That makes sense all right, particularly where you say 'The indefinite integral is the set of all antiderivatives'. I feel its a rather in depth question for the module I'm studying but what can you do!
cheers
Mikey

Not to be a smart arse but it's probably the most basic theoretical question one could ask in an integral calculus course!