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Use the Table of Integrals on Reference Pages 6-10 to evaluate the integral. $ \displaystyle \int \sec^5 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
Missouri State University
Campbell University
Harvey Mudd 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 evaluate the integral of C can't x to the fifth power. So to do this, we consult our formula for seeking to the end anti derivative. So all we gotta do is apply the formula. So if an equals five says that this is equal to tangent X c can't to the and minus two, divided by n minus one plus and minus two over and minus one times the integral of C can't to the n minus two d x. And then we can use the same formula again to analyze, seek and cubed vex. So in a girl of C can't cubed iss tan axe c can't of n minus two, which is just the first power over and minus one plus and minus two over and minus one times the integral of C can't to the n minus two. And we actually know the internal of C can't to the first power, which is Ellen of Seek an X plus Tangent acts. So now let's back substitute everything in. So our final integral of seek and to the fifth I'm just gonna do a little box over here because we need a lot of space tangent X c can't huge x over four plus 3/4 times are integral of C can't cubed, which is tan axe. Seek an axe over too, plus 1/2 Ellen of Seek an Axe plus Tangent X Plus C, of course. And if we do all the math and simplify our fractions here, we get the following plus 3/4 times. 1/2 is 3/8 and that 3/8 is also attached to our log term. And we have our integration constant so again only had to do was applied. The sea can't formula over and over again.
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