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Evaluate the integral.

$ \displaystyle \int_1^3 \frac{e^{\frac{3}{x}}}{x^2}\ dx $

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Calculus 2 / BC

Chapter 7

Techniques of Integration

Section 5

Strategy for Integration

Integration Techniques

Oregon State University

University of Nottingham

Idaho State University

Lectures

01:53

In mathematics, integration is one of the two main operations in calculus, with its inverse, differentiation, being the other. Given a function of a real variable, an antiderivative, integral, or integrand is the function's derivative, with respect to the variable of interest. The integrals of a function are the components of its antiderivative. The definite integral of a function from a to b is the area of the region in the xy-plane that lies between the graph of the function and the x-axis, above the x-axis, or below the x-axis. The indefinite integral of a function is an antiderivative of the function, and can be used to find the original function when given the derivative. The definite integral of a function is a single-valued function on a given interval. It can be computed by evaluating the definite integral of a function at every x in the domain of the function, then adding the results together.

27:53

In mathematics, a technique is a method or formula for solving a problem. Techniques are often used in mathematics, physics, economics, and computer science.

02:10

Evaluate the integral.…

01:03

Evaluate the definite inte…

02:45

Evaluate the given integra…

01:56

04:43

02:04

01:45

Evaluate the integrals.

01:57

0:00

02:53

03:30

Let's evaluate this inner girl by using to use up. They're probably several choices for you here that would work. For example, Let's just try one over X because if we look at do you negative one over x square D X and we see the X square down here on a DX over here. So looks like we're on the right track. So after doing this use of I'll first need a plot a minus sign that I got over here from the ISA. Then let's watch out for those limits of integration. So here X equals one. So plug it in and you get U equals one. But up here, plug it in for X U equals wonder. So that's our new upper limit. And then we have eat to the three over X dashes e to the one over X kun. So that's just here we have into the three U I guess. Yeah. Here, let me rewrite this. Make this a little easier to see. This is just e to the three times one of Rex. I should never end it that way. Let me write it this way. And that lets us, right? This is either the three you because this is just you here. And then I have d'You and recall that this negative with the deal is giving me the X squared on the bottom and then the DX up top. So go ahead and integrate this. If this three years bothering you, just do another use up here. And this will be your anti derivative plug in those that points. And then you get negative e to the one of all divided by three and then plus each of the three over three and we can go ahead and simplify this. Yeah, and that's your final answer.

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