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Use the Table of Integrals on Reference Pages 6-10 to evaluate the integral. $ \displaystyle \int \sqrt{e^{2x} - 1}\ dx $
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Calculus 2 / BC
Chapter 7
Techniques of Integration
Section 6
Integration Using Tables and Computer Algebra Systems
Integration Techniques
Missouri State University
Idaho State University
Boston College
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.
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Use the Table of Integrals…
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Use the Table of Integral…
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Okay, so this question wants us to integrate this function. So to do that, we need to transform it into one of the forms we can see in our table. So to do this, we always like to see a U squared insider square roots. So let's pick you to be equal to eat of the ex. So that means that do you is equal to eat of the ex DX. But we don't have that anywhere. So we need to divide by e to the X. So we get do you overeat of the ex is equal to D X So now we can transform. This is the integral of square root eat of the two x you squared minus one divided by archangel factor of e to the X Do you? But we can't keep this eat of the X in that form We need to change the u world now So either the axe is equal to you Then X equals the natural law give you so we can rewrite. This is the integral the square root of you squared minus one over e to the Ellen of you. Do you? Which is equal to well, he's to cancel, and this is an integral we can compute. We just look at the following formula from our table and in this case, a equals one. So that makes are integral square root. You squared minus one minus co sign in verse a over absolute belly of you plus c them plugging back in. We'll get our final answer of each of the two X minus one minus co. Sign in verse of one over absolute value of E to the X, which is just eat of the ex because either the exes always positive plus c giving us our final answer.
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