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Evaluate the given double integrals.$$\int_{-1}^{1} \int_{1}^{e^{x}} \frac{1}{x y} d y d x$$

2

Calculus 3

Chapter 29

Partial Derivatives and Double Integrals

Section 4

Double Integrals

Partial Derivatives

Missouri State University

Harvey Mudd College

Baylor University

University of Michigan - Ann Arbor

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we want to evaluate the double integral. That is the integral from negative 1 to 1 of the integral. From one to the V X. One over X Y. Dy dx. This question is challenge your understanding of integration standpoint of multi variable calculus. To solve. We're gonna proceed in two steps. In each step, we use single variable integration techniques where step one we compute the integral. That here is the internal from one E X of one over X. Y. Dy once you complete that were computing the outer integral to solve. So let's go ahead and proceed. The integral. Or rather the double integral is the integral negative 1 to 1 of one over X L N X Y E X to one dx. Or the integral negative 1 to 1 of L N X over X plus one minus Ln x over X dx. This was using long rules. We see the L n X over X cancels in this problem. So we're left with the integral from negative 1 to 1 of one dx. Thus our solution is simply X evaluated from negative 1211 minus negative one is one plus one equals two.

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