Expert: Socrates - 5/28/2005

Question
Hello, I have tried for hours to figure out how to integrate (evaluate) this problem, can you help me please?

the double integral from 2-4 and 4-5 (in that order) of ((x/y) + (y/x))dxdy...

Thank you ,
Brandon

Answer
An anti derivative for (x/y) + (y/x) with respect to x is (x^2/2y) + ylnx . Evaluate this at x=5 and x=4 and take the difference to get S (x/y) + (y/x) dx =
25/(2y) + yln5 - 16/(2y) - yln4 = 9/(2y) + yln(5/4).
An anti derivative for  9/(2y) + yln(5/4) with respect to y is (9/2)lny + (1/2)(y^2)ln(5/4). Evaluate this at y=4 and y=2 and take the difference to get
S 9/(2y) + yln(5/4) dy =
(9/2)ln4 + (1/2)(16)ln(5/4) - (9/2)ln2 - (1/2)(4)ln(5/4)=
9ln2 + 8ln(5/4) - (9/2)ln2 - 2ln(5/4)=
(9/2)ln2 + 6ln(5/4)
The answer is (9/2)ln2 + 6ln(5/4)
This can also be written as 6ln5 - (15/2)ln2 , using rules for logarithms, but either answer is correct.