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Use the Table of Integrals on Reference Pages 6-10 to evaluate the integral. $ \displaystyle \int \frac{dx}{\sqrt{1 - e^{2x}}} $
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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. This question wants us to integrate this function. Using a table so well, usually like, is having some sort of a you squared in the denominator. So let's have you equal eat of the ex. So that means that do you equals you to the x d x. But we don't have any of those. So do you. Over e to the X equals DX. So no, we can write this as d'you over either the X square roots one minus you squared. But we're still not done because we haven't eat of the ex and we're supposed to be in the u world. So if you is equal to eat of the X, then X is equal to Ellen of you. So now we can rewrite this as the integral of D'You over E to the Ellen of you Square group one minus you squared and we see the deed to the Ellen. You is just you. So we get our final integral of d U. Over you square root one minus you squared. And this is something that we can look up in our table. It's exactly one of the forms. So we get that this is equal to negative times. Ln of absolute value, one plus square root one minus You squared all over you plus C And now we can flip this fraction over and put them to get rid of the negative sign. So we get equals. Ellen of you over one plus square, one minus you squared plus c. And then the last thing we got to d'oh is plug in eat of the ex for you and now we're done.
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