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Problem 61 Medium Difficulty

Use Property 8 to estimate the value of the integral.

$ \displaystyle \int^{\pi/3}_{\pi/4} \tan x \,dx $


$\frac{\pi}{12} \leq \int_{\pi / 4}^{\pi / 3} \tan x d x \leq \frac{\sqrt{3} \pi}{12}$


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Video Transcript

problem. It is asked to use the property aid to estimate the value of the integral. Yes. Mhm dan ex d. It limit. Bye Bye. 4 2 five x 3 years. So we need to find the interior. Let us say this is I. That is invitation. So now we can write what we can. Right further I equals to nothing but this is this can brittleness. Bye bye forward to privately. That is limit same but tan X can be written as cynics have fun plus six problem. Right. So this is the it's now we can assume right we can as you right Cossacks as you let by Because cossacks into differentiation of differentiation of Cossacks is present on the upside right. So we can reply replacing this way. So this will become like this FDA shakes upon efforts so that that will give us the property like log effects. Right. Integration of that will be log effects. Let us see further. So let because eggs bu Right so it will be the differentiation of concepts will be nothing but this is minus cynics okay. Dx this is to you the questions you do. Great. So here we can also write cynics did because cynics did cynics T X is given to us which is equal to minus doing here sign it sticks can be replaced by minus the tax rate. So now therefore we will just put this value. This will give us leverage international do you buy deep you limit 542 vibrantly this is an of minus right minus taking from a near no this becomes worth. This becomes nothing but this is I question minus Ln you and then is the natural you know, So uh we know that you request to process so we can put the value. Okay that is Ellen was excellent right now maybe they applying the limit. Mhm mm Right, so I equals two. I will right here in all right, this is we have already integrated. So this is the integration. Right? She was her. Okay give us what this is. Give us I we will just put the limits. Right? So this should be limited to be like this. 542. Bye bye treat you Again. This is 542. Bye bye today here late. Mhm. So this is remember this is the value for X is supposed to this right? We are not changed to union by exchanging because we need to put the value again uh of you in terms of X. So that's why I have not changed that limit. So but in case if you don't change that limit then you have to change the limits according to you. Right? So this will be I. E. Question nothing but this will be minus and then course I begged me minus Ellen horse find me according to great. So that's why we gave us I equals two. Mhm Right, okay-. And then caused by Red three is nothing but This is to keep our manager or you can see one x 2. Okay. And this will be plus no. Cool to the power minus when? Right two. Mhm Yeah, I'm sorry. This becomes plus here. Great. Plus n in cause 54. Right? Because minus and minus will become plus here. So here it with me. We will be getting I equals two. Ellen too. Why? Because -1 is taken outside. We know the property of the no minus half. No two drink. So this will finally give us I'm sorry. This is again this is Ellen right? It should be pregnant. This will give us I equals two. You can save 5453. Mechanics the IDs is equals two and then to buy. Yeah, still in here. Mhm. Okay, so this is our answer to this problem. I hope you understood the concept. Thank you for watching. Yeah.