Evaluate each double integral.

$$\int_{1}^{4} \int_{x}^{x^{2}} \frac{1}{y} d y d$$

$4 \ln 4-3$

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Campbell University

Baylor University

University of Michigan - Ann Arbor

Boston College

Okay, here we have the girls here have one. No. One. One, two, four And in the inner and a girl X to X squared. Wanted for Why in the subsidy like Deke X already. So we have one for this is gonna be log Absolute value of why? From X to explain. Okay, so we have one, two, four log absolute value. Now we can actually drop That's of value here. I mean, of course, X squared is always going to be positive, but because we're going to be integrating from one to four, and that's positive. We're really just looking at log Why? And said This is going to be log of X squared minus log of eggs the ex. But there's something we can actually do to simplify this so long of X squared minus log eggs. Well Oh, we're actually so we're away from, say, I mean, this is just tripping. Protects between one for log of X squared minus log of X is log of X squared over X. So that's just before log of X X. And now we need to integrate log. And if you forget how to do this, use integration by parts. I'm just going Tio Hi news. The fact that I know is true that the anti derivative of Log X is, uh, city ex log X minus X And again, if you forget where that comes from, you just do the integration by parts he let you be Log X and Deedee. Pretty ex. Okay, so this is for log for minus for minus. Log one zero and then minus. Ah one. See? Right minus. But then minus a plus one, it's to be a four log for Katniss.

Georgia Southern University