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Numerade Educator



Problem 66 Hard Difficulty

Evaluate the integral.

$ \displaystyle \int_{\frac{\pi}{4}}^{\frac{\pi}{3}} \frac{\ln (\tan x)}{\sin x \cos x}\ dx $


$\frac{(\ln \sqrt{3})^{2}}{2}$


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Video Transcript

Let's use the use of for this one. Let's take you to be tan of X, do you? We know that is seek and square. And then we could go ahead and use the fact that C can't squared. Is tan squared plus one. So then that's a path, a grand identity. And then go ahead and replace tangent back with you and then solve this equation over here for the X. Oh, excuse me. Not that I circled around one and this one so soft for D s. So you get do you over use where plus one equals DX and we should also rewrite this. So let's look att We have won over sign eggs co sign X We go ahead and multiply top and bottom by co sign and then rewrite. This is one over Tan X and their c can squared x So I have won over you and then you swear plus one. So now let's go ahead and rewrite this integral after doing the use up so the end points will change. So go ahead and plug in pira for into disfunction up here the use of so plug it into the tan. So lower limit upper limit pie over three inside the town. And then we have natural lot of you. And then we just evaluated this over here, and it was equal to this. So we can do times you square plus one over you. But then we also have to multiply. We have to replace the X with this term over here. So our expression right now in the inside grand is this thing right here and then times do you over you squared plus one. And then here you would go ahead and cancel those terms they use clear plus one terms. So at this moment, after we cancel those herbs Well, Stan of pile before that's one and then tan a pie over three. That's rule three. And then we just have Ellen ofyou over you deal. So I'm running out of room here. Let me go to the next page one room three. Ellen, you over you. Do you? And we're almost unless she's due a use up here, then we can rewrite. This is change that limit of integration. That's Ellen one Ellen, Route three. And then we just have W d w so w squared over two. That first and point here is zero Elena born and then l in room three. And so plug that in. We get Ellen, Route three, that's square, and that's divided by two and then minus zero, So that's a finalist.