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Use a computer algebra system to evaluate the integral. Compare the answer with the result of using tables. If the answers are not the same, show that they are equivalent.

$ \displaystyle \int \tan^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

Harvey Mudd College

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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Use a computer algebra sy…

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Use a computer algebra sys…

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Okay, So this question wants us to evaluate this integral using technology and show that its form is equivalent to that given by the table. So a computer algebra system gives us this anti derivative. Well, the table gives us this one says you can see the tangent to the fourth term and the change into the second term both match. So we just need to show that these 2/3 terms are equivalent. So I need to show that negative Ellen of C can't axe equals hello and of co sign X. Okay, So this is pretty straightforward to dio because consider negative alone of seeking ex. So is the same thing is negative, Ellen of well, c can't is one over co sign X. So that means it's negative. Ellen of co sign X all raised to the negative first power end by lug properties. We can bring this exponents outfront and get negative one times negative Ln of coastline X and that simplifies to just Ellen of co sign X. So we've shown that negative Ellen of C Can X is the same thing. Is Ellen of co sign X So we're done because that means that our table and computer algebra system answers are absolutely the same

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