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Evaluate the integral.

$ \displaystyle \int \tan^2 x \sec x dx $

$$

\int \tan ^{2} x \sec x d x=\frac{1}{2} \sec x \tan x-\frac{1}{2} \ln |\sec x+\tan x|+c

$$

Integration Techniques

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Campbell University

Oregon State University

Harvey Mudd College

Boston College

this problem is from Chapter seven section to problem number thirty two in the book Calculus Early Transcendental lt's a Condition by James Store. Here we have a indefinite general of tangent square time seeking. The first thing we can do here is applied this but Agron identity on the right to rewrite tension square as seek and squared minus one. So that's Tangent Square. And then we still have seeking on on the right. So let's go ahead and and break this up into two in a girls. So we have after we distribute to seek and through the apprentices seeking Cube, let me switch colors here and the worlds he can't. So it will take some work to evaluate these interval separately. So let's do note this first integral here buy some letter is called I and for the second integral listing All this by J. So let's deal with first. So I One way to go about this is to try integration by parts so we could take you two be seeking then do you? Is CNN, Times, tangent GX and we have amy sequence where D X And the reason for doing this is because then we know. If devious, seek and square. Then we know that V is simply Tanox. Softer playing integration My parts we have that eyes. U V minus integral me to you. So we have seek in next time Stan Genetics minus and a girl of V do so tangent time seeking times tangent some tangent Square Lex Time Sienna Brooks, D X And once again, here for attention Square. We can apply this. I got you on identity to write that as sequence Where? X minus one. So this's all equal to seeking of eggs. Time's tangent of eggs minus now we have seek and square times he can't That's seeking Cube, which you also will note that this is our original integral I That's a good thing. And then we have two minuses here, minus minus. So that's a plus in a girl of seeking, which you also notices are R. J. So since we'LL evaluate J eventually, let's not go ahead and evaluate the integral seeking for right now. So since we have eye on the left and I agree on the right from this latest equation, we conclude that we have two. I equals seeking a books Tangent of X Plus C convicts Integral Seeking a Vicks Dix equivalently Let's divide by two plus Integral Seek and over too and again we're not. We don't need to simplified this integral yet because what we're going to have to subtract J anyways. And that's also then we'LL seek. And so let's know that right now when we do, I'm minus J, which is our answer. This is the thing that we want. This is what our are integral became. I'm minus share. We have seeking times tangent all over, too. Then we have integral C can over to. But then when we subtract, J were subtracting integral seeking so that one half minus one will give us a minus one half and a roll seeking. So now we'LL have to go to the side and evaluate the integral Seek Innovex. So let's do that. So one way to proceed for the integral seeking of X. So let's ignore the dividing by two. For now, we'LL deal with that. At the very end. One way to proceed is to multiply top and bottom by the same term and this taste in this case we can do this by seek in a box plus tan genetics. So we haven't changed the interval because we we just multiplied. Seek and buy one. So it's good and multiply out that numerator by distributing the Sikh. Enter the numerator they have seek and squared X plus Eka. Next time San genetics and the denominator just have see camp list and and observe here we can apply u substitution. Let's take you to be the denominator Then do you becomes the numerator? You're a little See Candace See Ken Times tangent It's apprentices and then the derivative of tension A sequence where DX which is exactly on so D'You is our numerator as usar denominator. So plugging this in we have won over you, do you which is natural log of absolute value of you and we can replace back substitute by replacing you with seek and plus tangent. So going back to our previous expression for I mine is Jane, we have seek Innovex Time stands Innovex over too. And then we had minus the interval of seeking over too. So minus Ellen C can't plus tangent all over too. And we could do minus C over to. But since he's just a constant. It doesn't matter what letter we use here. If we multiply, divide, see by a number of changes, sign. So if we want, we can go ahead and just put plus here on the right, and there's our final answer.