Evaluate each iterated integral. (Many of these use results from Exercises $1-10 .$

$$

\int_{1}^{3} \int_{1}^{3} \frac{1}{x y} d y d x

$$

$(\ln 3)^{2}$

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Okay, so we're giving this integral, um, one of three times enable from 1 to 3 of the function, one over x times Y de y dx. Okay. So, first, what I'm gonna do is I'm just gonna rewrite this inside a little bit. So it's actually just one times X and at one times why. And then we're gonna evaluate the inside intervals. So 123 of one times X one over X and that one over. Why de y Andi? Since we're integrating respect to lifers, we're going to treat every other Grable, Wisconsin. So this allows us to move the one over X outside, and then we have the new road from 1 to 3 of one over. Why d y valuing this? We have three anti derivative of one over by Is Ellen of the absolute value of why rare? Don't forget the absolute values. They're very important. But for this problem, not too much. Uh, but silver murdered to put them in. So we're valuing this from 13 and now plugging this end, we have Ellen of all civilian three highest Ln of absolutely one. And the absolute values aren't too important. This time because both our numbers are positive. So you don't really matter too much. And this should result in we use the property of logs. We have the Ln of 3/1, which is just the same as one of her axe times, Ellen of three. And then we're gonna just drop these absolute value bars because the OB civility three is three. So, Ellen of three. So this is the result of the inside and roll. So now we're just gonna move to evaluate the outside a nickel. So we have the interval from one of three of I was going to move the Ellen three to the outside because it's a constant one over x dx. This is equal to Elena of three times Ellen of the stops, the value of X, evaluated from 13 And knowing that this resulted in Allen three, um, this should result in Ellen three as well, because they're basically the same. It's just like a different alias we have. Why here and acts here. But we're basically playing in the same covers. Three. So this should definitely result in the same thing. So, Eleanor, three times Eleanor three. So we have Ellen of 3 20 square, and that is our final answer.

Rutgers, The State University of New Jersey