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Evaluate the integral.

$ \displaystyle \int \tan^3 x \sec x dx $

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Calculus 2 / BC

Chapter 7

Techniques of Integration

Section 2

Trigonometric Integrals

Integration Techniques

Campbell University

Harvey Mudd College

University of Michigan - Ann Arbor

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.

01:55

01:35

Evaluate the integral.…

01:22

00:59

01:03

Evaluate the integrals.

01:08

Evaluate the integrals…

this problem is from Chapter seven section to problem number twenty seven in the book Calculus Early Transcendental lt's a Tradition by James Store We haven't indefinite inaugural of Tangent Cube time, so you can't since the power on seek an Izod is just one here. Let's pull out a factor of tangent times he can outside of this inner grant. So I have tangent Squared X And then here's the other power of tangent Times are seeking and the reason for doing so is that we'll eventually be able to deal with this term tangent time c can when we do it u substitution and that you saw it should be u equals seek an ibex But before we do that, we'Ll have to deal with this tan squared over here on the left So let's rewrite this tan squared Soto, rewrite this tan square We can use our identity Our protagonist identity on the right Over here circled c can square is tan squared plus one So that means town square is C can't squared minus one So that becomes That's our chance Weird and we still have this remaining term left over. Now we see that we have the right choice for our U substitution. Let's take you to be seeking so that do you is seek Innovex Time's tangent of X dx. So if we come backto are integral we see that here we have in terms of our use of we have a use squared minus one and the remaining term over here is our do you So integral becomes you squared minus one. Do you? And we can evaluate each of these inaugurals by using the power rule. So you swear becomes you, keep over three and one just becomes you and lets out our constancy of integration. And at this point, we can We're done with the integration. But we should put our final answer in terms of X. So we come back to the U substitution to rewrite this. And so you cute becomes seeking kun the Vex mine issue, which is just seeking a vex plus C. And there's our answer

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