• Support PF! Buy your school textbooks, materials and every day products Here!

How Can I prove this?

  • Thread starter S_klogW
  • Start date
  • #1
19
0

Homework Statement


Let's definite the function f(x)=∫(from x to x+1)sin(t^2)dt



Homework Equations


There is another function of x:
g(x)=cos(x^2)/2x-cos((x+1)^2)/2(x+1)


The Attempt at a Solution


Prove that when x→+∞,there is the equation:
f(x)=g(x)+O(1/(x^2))
Here the O(u) means that when u→0, the O(u) is at least infinite small comparable to the infinite small quantity u.

I am only a 12 grade highschool student, so I have no methods to solve this. I will be grateful if you could give me some advice. This is a problem from the exercises of the Mathematical Analysis by V.A.Zorich, chapter 6, Volume I.
 
Last edited:

Answers and Replies

  • #2
19
0
Ha, I have solved this question. I got this problem from the Chinese Edition of the Zorich Analysis and it has a typo so I could not prove this. The problem in the English Edition is correct.

Consider ((cos(t^2))/t)'=-2sin(t^2)-cos(t^2)/(t^2)
The second part of the right side could be estimated smaller than 1/(x^2). So the problem solved.
 
  • #3
hunt_mat
Homework Helper
1,739
18
Write:

[tex]
\cos x^{2}=\frac{2x\cos x^{2}}{2x}
[/tex]

and use integration by parts.
 
  • #4
19
0
Write:

[tex]
\cos x^{2}=\frac{2x\cos x^{2}}{2x}
[/tex]

and use integration by parts.
Yes, I have already known that.
I could not solve it because there is a typo in my Chinese Edition of the book.
 

Related Threads on How Can I prove this?

  • Last Post
Replies
9
Views
2K
Replies
6
Views
2K
  • Last Post
Replies
3
Views
1K
  • Last Post
Replies
4
Views
1K
Replies
1
Views
764
Replies
4
Views
2K
Replies
1
Views
236
Replies
2
Views
1K
Top