How to prove the definition of arctangent function using integral?

[zhongbeyond]

Homework Statement

This is a problem from Introduction to Analysis by Arthur P. Mattuck,chapter 20,problem 20-1.

<a href="http://www.flickr.com/photos/86024731@N04/8090259684/" title="arctangent by gnu is not unix, on Flickr"><img src="http://farm9.staticflickr.com/8193/8090259684_a5ce06801e.jpg" width="500" height="325" alt="arctangent"></a>

I have worked out question (a),(b),(c),but I am stucked in question (d).
Please help,thanks.

 

[jambaugh]

For d.) you will need to compare the given integral to the integral of some other function which you can show (easily) has limiting value < or = to 2.5 at infinity.
Suggestion: consider changing the numerator to make it integrable. Be sure to check that your comparison function is everywhere bigger. (You may need to use some glue.)