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Use the Table of Integrals on Reference Pages 6-10 to evaluate the integral. $ \displaystyle \int \frac{\sqrt{4 + (\ln x)^2}}{x}\ dx $

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Calculus 2 / BC

Chapter 7

Techniques of Integration

Section 6

Integration Using Tables and Computer Algebra Systems

Integration Techniques

Campbell University

Harvey Mudd College

Baylor University

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.

03:06

Use the Table of Integrals…

03:15

02:06

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03:50

Use the Table of Integral…

05:43

02:40

04:01

07:55

02:16

04:20

Okay, This question gives us this end girl to evaluate. So to do this, we're gonna want to put it into a nice form that we can look up in the table so that Ellen axes what's causing problems, So let you equal Ln of eggs. So that means that do you pickles one over x d x, which conveniently enough we have in our integral. So this means are integral becomes the integral of square root four plus u squared. Do you? And we can fix this integral sign a little bit. Okay, so now from there, there's a very familiar in circle based on our table. So we see the following formula, and in this case, a A's equal to his two squared is for so are integral becomes you over too square root four plus u squared plus four over too Ellen of you plus square root, a squared plus u squared plus c. But we could replace this a squared with before and now all we have to do is we can make one simplification with this four over too. And we can just plug in Alan of X in for you. So we get Ln of X over too Times Square Root four plus Ellen of X squared plus to Ln both Ellen X plus Square root for Plus Ellen of X Squared plus c. So we got some nested longer of them's here, but being extra careful with everything, this is our final anti derivative.

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