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Use the Table of Integrals on Reference Pages 6-10 to evaluate the integral. $ \displaystyle \int \frac{e^{3t}}{\sqrt{e^{2t} - 1}}\ dt $

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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

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.

03:33

Use the Table of Integrals…

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Use the Table of Integral…

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Use the formula in the ind…

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02:48

Okay, This question wants us to evaluate this integral using our table. So to do that, we need to get this into a form that our table can recognize. So we see we have a square root in the bottom and usually with square roots, we like a you squared in there somewhere. So let's call each of the two t you squared. So if we square root both sides, that means that you is equal to eat a t. So d'you is equal to either the t d t. Then we can rewrite this integral so we can see what's going on here. So we get the integral of each of the two tee times e to the t d t. We're just splitting up the top here times the square root of eating the two tea minus one. And we know that each of the T d t is, do you? And then the each of the two teas that are left just become you squared. So we have a use squared times a d. U over square roots of you squared minus one. And now this is in the form that our table recognizes. So we see that this integral evaluates to the following expression. This should be a minus. Sign here, though. Sorry. And now we have to do is plug in you and plug in a So a squared is one. And to you is e to the t So plugging that in we get eating that sea over two times the square root of each of the two tea, minus one plus 1/2 times the natural log of either that C plus square root of B to the two t minus one plus c and that's our final answer.

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