Proving Function Equations: f(x) and g(x) with O(1/(x^2))

[S_klogW]

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 high school 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.

 

[S_klogW]

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.

 

[hunt_mat]

Write:

$$\cos x^{2}=\frac{2x\cos x^{2}}{2x}$$

and use integration by parts.

 

[S_klogW]

Write:

$$\cos x^{2}=\frac{2x\cos x^{2}}{2x}$$

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.