# Simple indefinite integral question

Discussion in 'Math' started by MichealY, May 23, 2009.

1. ### MichealY Thread Starter Active Member

Apr 9, 2009
49
0
$$\int cost^2dt$$ = ?

Thanks.
MichealY.

2. ### Ratch New Member

Mar 20, 2007
1,068
3
MichealY,

What are you trying to integrate? cos^2 (t)dt, cos(t^2), or what? Let's use some parentheses so we know what you what you are talking about. Can't you look that up in a table of integrals?

Ratch

3. ### MichealY Thread Starter Active Member

Apr 9, 2009
49
0
Ratch:
It is cos(t^2) not cos^2(t).
No I cannot look up in a table of integrals.
I want to know how to solve it by us,altough it could be solved by MATLAB/Mathematica.

MichealY

4. ### Nukular New Member

May 16, 2009
3
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Change variables from t^2 to x, then integrate by parts?

Last edited: May 23, 2009
5. ### MichealY Thread Starter Active Member

Apr 9, 2009
49
0
Nukular:
What do you mean by integrating by parts?

MichealY

6. ### t_n_k AAC Fanatic!

Mar 6, 2009
5,448
782
Presumably you realize the solution is an infinite series.

7. ### MichealY Thread Starter Active Member

Apr 9, 2009
49
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t n k,
Bingo.I asked someone of mathematic major the other day.He said it is a problem of infinite series,But I do not even know what's infinite series,so I need the help of you,and at the same time I will work hard to learn infinite series

MichealY

8. ### t_n_k AAC Fanatic!

Mar 6, 2009
5,448
782
Hi MichealY,

As Nukular rightly points out the solution probably requires an understanding of both substitution of variables and integration by parts. It's the approach I would use anyway. Your earlier post suggests you are not familiar with the latter method. It would be useful to do some background study on that first and then return to this problem. Are you studying integration as part of formal course or is this something simply of personal interest?

9. ### MichealY Thread Starter Active Member

Apr 9, 2009
49
0
Nukular,
I have understood a bit of what you mean.

t n k,
Yes, I am not familiar with integration by parts,maybe I just don't know this English phrase I will do some background study and then return to this problem.I'm studying integration as part of formal course(I failed the final exam,now trying to make up),and it is something of personal interest too.

Thanks to both of you.

MichealY

10. ### Mark44 Well-Known Member

Nov 26, 2007
626
1
Here is integration by parts in mathematics notation:
$\int u dv = uv - \int v du$

It is the product rule of differentiation in reverse.
The trick to using this technique is to identify what part of the original integrand is u, which the rest being called dv. From this you antidifferentiate dv to get v, and take the differential of u to get du. If you have chosen u and dv correctly, the new integral will be easier to find than the one you started with.

11. ### MichealY Thread Starter Active Member

Apr 9, 2009
49
0
Mark44,
Thanks.I have had learnt integration by parts before.I thought it means,first,reduce cost^2 to many parts,then integrate them by parts.

I have also tried this method,but it seems that it won't work.

MichealY

12. ### t_n_k AAC Fanatic!

Mar 6, 2009
5,448
782
It does work.

13. ### t_n_k AAC Fanatic!

Mar 6, 2009
5,448
782
Recognizing that the solution is an infinite series, the other approach would be to expand cos(t^2) as an infinite series and integrate that wrt t. The trouble with any solution one obtains using any approach is getting the answer into an agreed "standard form" - i.e. as agreed by mathematicians. I haven't looked up a standard handbook of integral tables for this function, but I would think the agreed solution has long been cast in stone. Finding a derivation might be more difficult - I guess it's out there somewhere!

One could probably better spend time doing practice on any number of functions with finite solutions of increasing degree of difficulty.

Mar 6, 2009
5,448
782

Nov 9, 2007
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16. ### t_n_k AAC Fanatic!

Mar 6, 2009
5,448
782
Good find Studiot! - hopefully MichealY will find this helpful.

17. ### MichealY Thread Starter Active Member

Apr 9, 2009
49
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Thanks,you are really kind.

I mean I have tried method of integrating by parts.

And by using MATLAB,the result returned contain Fersnel C.

MichealY

Last edited: May 27, 2009
18. ### rin-elt New Member

Jun 18, 2009
9
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Hi I also do think this is related to infinite series
try and use sandwicth theorem to find whether it is convergent and find it's limit,if i'm not mistaken this limit is the value of the integral

19. ### MichealY Thread Starter Active Member

Apr 9, 2009
49
0
Starting to study numerical analysis,I have learnt a bit
It seems that langarage polynomail is better than taylor polynomial.