# Antiderivative of a fraction

Discussion in 'Math' started by jaygatsby, Apr 4, 2012.

1. ### jaygatsby Thread Starter New Member

Nov 23, 2011
185
23
Hello -

I'd like to take the antiderivative of this function: (x/2 + 3) dx.
However I don't know how to handle the x/2 part! Hints?

Thank you,
J

2. ### steveb Senior Member

Jul 3, 2008
2,433
469
Integral tables are available to help you when you forget, or when the function is not simple. This is one you should know by heart though.

$ax^n$ has antiderivative of ${{ax^{(n+1)}\over{(n+1)}}$ when $a$ and $n$ are constants.

3. ### amilton542 Active Member

Nov 13, 2010
496
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It proves useful to memorise the tables of derivatives and anti-derivatives on a scrap piece of paper in your pocket.

4. ### Georacer Moderator

Nov 25, 2009
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1,266
Why did you call it anti-derivative instead of integral? Is there a difference between them in English?

5. ### amilton542 Active Member

Nov 13, 2010
496
64
I've never heard of someone refer to an integral as an anti-derivative in England.

Once I realised there was a limit on how deep my college teach calculus, I turned to the MIT videos on single variable calculus to fill in the gaps.

The terminology 'anti-derivative' was used so excessively in these videos, the nomenclature just washed-off on me.

In all fairness, I prefer using 'anti-derivative' instead of 'the integral of.'

6. ### steveb Senior Member

Jul 3, 2008
2,433
469
They are so similar they are essentially the same. However, integrals usually have assigned limits, while the antiderivative is like the indefinite integral and without implied limits.

By the way, that reminds me, I forgot to include the constant. Antiderivatives are only specified within an additive constant, so always include a +C at the end of one.

7. ### amilton542 Active Member

Nov 13, 2010
496
64
@ steveb

Is this a bad habit to refer to an integral as an anti-derivative?

I even refer to a definite integral as an anti-derivative.

8. ### steveb Senior Member

Jul 3, 2008
2,433
469
I think anti-derivative and indefinite integral are effectively synonymous, - or at least they are to me. Hence, in the context of this thread, I think your language is correct.

Perhaps it is a bad habit to call a definite integral an antiderivative. If you prefer to have one term that covers it all, I guess "integral" is the better choice. IMHO

9. ### amilton542 Active Member

Nov 13, 2010
496
64
The terminology 'anti-derivative' does have a nice ring to it. I think I've got a bit carried away with it. From now on, it stays with the tables only.

10. ### russ_hensel Distinguished Member

Jan 11, 2009
821
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the fact that the 'anti-derivative is equal to the derivative is so important that it is called the "Fundamental theorem of Calculus". Wow!

11. ### amilton542 Active Member

Nov 13, 2010
496
64
So what are you trying to say, the nomenclature 'anti-derivative' can be applied to definite integrals as well as indefinite integrals, yes or no?

12. ### steveb Senior Member

Jul 3, 2008
2,433
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I think what he is trying to say is that the great discovery of both Newton and Leibniz is that definite integrals (which are areas under a function) can be calculated using the antiderivative of the function evaluated at the endpoints. Before that discovery, anti-derivative and indefinite integral were not synonymous.

And, yes, I agree, - that is a big WOW! That basic concept leads to the generalized Stokes' Theorem, where an integral of a function over a boundary is equated to the integral of the exterior derivative inside the boundary. The fact that Maxwell's electromagnetic equations are just 4 statements of the generalize Stokes' Theorem shows how important it is to us in this forum.

I don't think this implies that the terminology "anti-derivative" is the same as "definite integral", strictly. But, clearly they are very closely related. Evaluation of definite integrals (which are really just numbers in the end) is done using the indefinite integral (also called the antiderivative).

amilton542 likes this.
13. ### Georacer Moderator

Nov 25, 2009
5,151
1,266
I see. In Greece we only use the terms definite and indefinite integral. My confusion stemmed from that fact.

Thanks.

14. ### russ_hensel Distinguished Member

Jan 11, 2009
821
49
Shame on all of you for not catching my "typeo" ==> the fact that the 'anti-derivative is equal to the derivative is so important that it is called the "Fundamental theorem of Calculus". Wow! <== that second occurrence of derivative should be integral thus : The fact that the 'anti-derivative is equal to the integral is so important that it is called the "Fundamental theorem of Calculus". Wow! Even more shame on me for not catching it myself in the first place.

Last edited: Apr 7, 2012
15. ### steveb Senior Member

Jul 3, 2008
2,433
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We saw it, but your meaning was clear, so it didn't seem worth pointing out. We all mis-speak from time to time.

Anyway, that's a good point you make about the importance of the Fundamental Theorem of calculus.