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Evaluate the integral.

$ \displaystyle \int (1 + \tan x)^2 \sec x\ dx $

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${2sec} x+\left(\frac{1}{2} \sec x \cdot \tan x+\frac{1}{2} \ln |\sec x+\tan x|\right)^+C$

Calculus 2 / BC

Chapter 7

Techniques of Integration

Section 5

Strategy for Integration

Integration Techniques

Campbell University

University of Michigan - Ann Arbor

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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09:32

02:19

Evaluate the integral.…

01:25

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00:40

05:58

00:32

Evaluate the indefinite in…

04:59

00:45

So the first thing we can do here is just to go ahead and multiply the one plus tangent by itself. So we get one plus who tan plus stance, where all times he can't and then go ahead and multiply out the sea can. So, after multiplying, seek and this is what we have and recall that we can rewrite tension square as she can't square X minus one. This is one of your put that green identities. So go ahead and multiply seeking through the parentheses to each of these two. What? After multiplying that out, we have seeking Q Becks minus he can ex. And you could see from here that we could cross off some of the sea cans, cross this with the negative over here, and then we could go ahead and split this into two intervals that we can do. And then here we have a plus in a girl seeking cute. Yeah. Now over here for this first integral. We know that this is just two times the seeking of X. The revival of Sikkim is tangent, and he can't and then for the second and we're over here. This one is a little bit more. A bit more time to me. You can do pie parts here. So for buy parts, you could take you to be seeking so that do you equals C can't type tension. And then you would take Devi to be sequence where so that v equals ten X So you would use the integration by parts here. Well, in that formula, C really write the integral with UV minus an unworldly do you and then simplify that and that whole term becomes so now in this red color, one half see time stands in and then also one half natural on absolute value. She can't plus Tanja. And then let's add that constant C at the variant. And that's our final answer. And again the second integral here we get by using these values for you and Dee used the integration by parts, plug it into the formula up here and then simplify bee in general. And this is our term down here and read. And that's your final answer

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