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Evaluate the integrals.$\int \sec ^{4} x \tan ^{2} x d x$

$\frac{1}{5} \tan ^{5} x+\frac{1}{3} \tan ^{3} x+C$

Calculus 1 / AB

Calculus 2 / BC

Chapter 8

Techniques of Integration

Section 3

Trigonometric Integrals

Integration

Integration Techniques

Campbell University

Baylor University

University of Nottingham

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This is from fairy age from chapter 8.3. We have seconds to the fourth x times changing square x t x So here, when they were. So this kind of internal is a special family off intercourse it on Lee or seconds and change it functions and they're usually one of two substitution. Sze usually work so the 1st 1 it's so so either we do you in course second off. No Max, Are you seacoast? Ancient off axe One of the substitution Sze will usually work and let's see which one would work. So if you're not sure what to dio which one is the crest one right away So in this the 1st 1 we have this's second of exchange int of x t x So if you look at this that we have So this implies that our interim function will become you Cube And then there's one power of tension left and so that is going to be a problem because tangent off X again is equal to by definition Well, it's squared the U squared minus one. No! So in this substitution, we will deal with descend around. So while this isn't the worst into a weird thing. This this's over something that Khun Dubai intuited my parts several times. But that's what happens in the second substitution. In this case, we will have do you or bees will be just second squared X the X So in this case, our interim will become, um since so would see So I will have our 10. So we have second to afford extension squared off axe DX Here it's here and this is going to be a call to Well, if you in course tension and change of squares simply you You squared that two powers off the second who simply become tea, do you and then what's left? Here's tension squared of X But this is good because we have tension square. So it is simply you squared times you squared minus one. Do you so hear this thiss is obviously the better idea. So it's this proceeded with this one. I'm going to cross off this here so this will only become simply into a future for you my city of second to you that a suitor fish are fished this year, Harry. Third plus C there is equal to home so Now he's back. Substitute, sir. It's simply attention tw to her first over five, my ass tangent to certain over. Very close. See? Now, let me finish is probably saying this This is very similar to problem through six of this chapter where we have our powers of seconds intentions. So as you've seen and here we have even powers here both, uh, our foreign part two. And in this case, this substitution Rico's tension ex works out the best while in problem 36. If you chicken out who we will see, there's odd powers for both and in that case, the other substitution works the best.

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