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Evaluate the integral.
$ \displaystyle \int \frac{dx}{\cos x - 1} $
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Calculus 2 / BC
Chapter 7
Techniques of Integration
Section 2
Trigonometric Integrals
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.
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Evaluate the integral.…
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here we have the integral of one over co sign of X minus one. One way to proceed here is to multiply. That happened the bottom by one plus co sign X. So, after doing so, let's go ahead and multiply out the denominator with co sign X plus one in the numerator and in the denominator, we have cosign squared X minus one. So using the Pythagorean identity, we can rewrite this denominator as negative science. Where and then let's go ahead and split up this into grand into two fractions. We have coastline, eggs, overnegative, sign square plus one over negative science where using our knowledge of trigonometry, we could go ahead and re write these. So for the first in our group, we have negative Cho attention of X time's cozy Kennebec and for the second integral, we have negative Kosi can square of x, d, d x and we we know from our table or from previous examples that evaluate each of these two. So this is cozy convicts and the integral of negative Kosi can squared is coast engine of X, plus our constancy. And that's our answer
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