Trig Substitution... How? (NancyPi)

MIT grad shows how to integrate using trigonometric substitution. To skip ahead: 1) For HOW TO KNOW WHICH trig substitution to use (sin, tan, or sec), skip to 0:18. 2) For how to do a SIN SUB and WHEN TO TRY TRIG SUB, skip to 1:29. 3) For a TAN SUB with the radical in the NUMERATOR, skip to 14:41. 4) For a SEC SUB with a RATIONAL/FRACTIONAL POWER, skip to 22:36. 5) For miscellaneous types, skip to time 24:54. Trig substitution integration is a calculus technique for integrals. Nancy formerly of MathBFF explains the steps.
For my video on how to do BASIC INTEGRATION, jump to: • Basic Integration... H...
For my video on how to integrate using U-SUBSTITUTION, jump to: • How to Integrate Using...
For my video on how to do INTEGRATION BY PARTS, jump to: • Integration by Parts.....
Follow Nancy on Instagram: / nancypi
Twitter: / nancypi
HOW TO KNOW WHICH trig substitution to use: If the x-expression under the radical has the form of "a number minus an x^2 term", then you'll use a SINE substitution. If it has the reversed form of "an x^2 term minus a number", then you'll use a SECANT substitution. And finally, if the expression under the root is addition (the form of "an x^2 term plus a number"), you'll use a TANGENT substitution for the problem.
WHEN to use TRIG SUB: If you have an integral with a radical expression in it, and you know you cannot integrate using: a Table of Integrals integration rule, or using U-Substitution, or using the Power Rule, or using other integration techniques like Integration by Parts or Partial Fractions, then it is a good idea to try trig substitution.
HOW TO DO TRIG SUB. Here are the main steps of trig sub problems: 1) Decide which trig substitution you need [x = asin(theta), x = atan(theta), or x = asec(theta)], filling in the value for "a". 2) Simplify the radical expression by plugging in your substitution for x in the radical [For example, for a sin sub, plug in for x in sqrt(a^2-x^2)]. 3) Find dx from your x substitution. 4) Substitute into the original integral, replacing the radical, the dx, and any other x factors, with the theta expressions you found. 5) Simplify and integrate, using whatever technique you have to use. 6) After you integrate, it's still in terms of theta. So the last step is to get it all back in terms of x. Draw a right triangle off to the side, label the sides with what you know from your trig substitution expression, find the third side, and use the sides of the triangle to get an expression for whatever trig function you need in your final answer.
For more of my calculus antiderivative, indefinite integral, definite integral, and derivative math videos, as well as more examples and help with integration rules, using an integral table, integration formulas, trig integrals, completing the square, and a review of right triangle trigonometry, check out: nancypi.com

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"I'll be right by your side...."Prof during final: "Who is this?"

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my youtube math tutor

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seems legit

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"This is, a bit of a clusterfuck. This is more than you would ever ask for." THANK YOU! SOMEONE SAID IT!

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7:51 Did she just say "cluster-fuck" i've never heard her use a cuss word before. amazing

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Cussing has its place, and this is one of them.

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Nah! She's too innocent to know any cuss words.

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I had to double check

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She says it so sweet and innocent

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That sound is strangely satisfying. Lol

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Me too!!!!

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Uk,,i love the ma sound,actually i also watch her videos coz of it😬

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Probably a drop of 3 in one oil would reduce that friction noise.😋

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Same

[@shadrackokello8228]

Hello Nancy , I'm Shadrack , from Kenya. Pursuing Bachelor of Education in Math & Business Studies as a high school teacher. I can confirm that , you have been so helpful in my math units. Especially Calculus 1 & 2, ODE , Basic Mathematics. You explain seamlessly with ease. Sending my love from Kenya 🇰🇪🇰🇪🇰🇪

[@battlecoder]

Trig substitution was definitely the trickiest thing for me. With your explanations however it all makes sense. You are a superb teacher. Thanks.

[@xtratv8354]

"I know this is a bit of a cluster-f*ck" ICONIC and SAVAGE

[@xtratv8354]

7:50

[@icebearreal]

@@xtratv8354 Thanks m8

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@Caden Smith Imagine having a name like "Caden". Ya that's right, I'm picking a fight on a calculus tutorial.

[@sneekysmurf2084]

its comments like this that honestly really help me learning! having a laugh during a lesson makes it a MILLION times more valuable to me.

[@allengillean849]

Yea, this made me love her that much more lol

[@MrVoayer]

OK ! Let's be honest: Trig substitution is a difficult stuff. There are a lot of rules, a lot formulas to know by heart, a lot of tinkering around... But you certainly did a great job in covering a lot of it in a really understandable and helpful manner. This video should be seen several times until it is all sinks in !

[@mmghuneim]

Hey Nancy, I would love it if you did a video on series, and different tests that you use (Divergence Test, Ratio Test, etc.) to find the convergence or divergence. That's my main struggle.

[@Squash101]

Not even capping a tad bit this is by FAR the best tutorial I've seen. Jesus this is amazing thank you SO MUCH

[@dananajj]

7:51 excuse me? Did I hear right?

[@cometmace]

Turn on captions. You heard what I did. This is reality math.

[@Tethloach1]

yep you sure did.

[@szejer4184]

Honestly though. Ive described Trig sub in exactly those terms after looking at an entire white board covered after solving a single problem...Totally appropriate. :D

[@DecafMathASMR]

hahaha

[@kevinbush4300]

So glad your comment is the top thread. I don't have time today to check if anyone else thought the same. 😂

[@celineceline3141]

thank you for all your vids, nancy! i have just finished this last from all your other videos as my refresher before i start my computer science bachelor’s degree. i definitely got mindfucked with the clusterfuck and trig substitution as a whole, but i feel i got prepped and pumped well with your help! i hope i can still ace in the university given that i took a 2-year leave of absence because of health reasons. i wish you well, nancy! looking forward to your comeback 🤞🏼

[@AxOutdoors]

I feel lost knowing that this is as far as Nancy takes us into calculus 😭 thank you Nancy!

[@davisbeauvais6965]

im waiting for the integration by partial fraction decomposition video that never came :(

[@adnanmohammad1960]

Wish me best of luck guys. It's midnight now & tomorrow is my exam. But here I am binge watching her tutorials which are not related to my exam topics at all.

[@SoupyDoopieFan]

This channel has literally the ONLY example I could find with a coefficient next to x. You're a lifesaver!

[@Prof_Jeff]

You do a very good job. I recommend your videos to students who need supplemental material outside their textbook. I do want to point out two things: 1) Starting around 3:56 when you go through the algebra of the substitution x=6*sin(theta), while that is reasonable to do in order to re-enforce WHY x=a*sin(theta) works in this situation, if you merely draw the right triangle that corresponds to the relationship x=6*sin(theta) at this point, elementary trigonometry yields sqrt(36-x^2)=6*cos(theta) without having to talk about why we assume cos(theta)>0 in the algebraic approach (this often confuses students if it even is mentioned that technically sqrt(36-x^2)=6|cos(theta)| here). Constructing the right triangle from x=a*sin(theta) both helps the visual learner and avoids the awkward algebra, or at least the explanation of why absolute value is not employed. 2) At 11:05 when you want to start the back-substitution, you have all the information needed since you obtained a substitution for the radical expression algebraically: simply solve the equation sqrt(36-x^2)=6cos(theta) for cos(theta). Also, if you construct the right triangle first (as suggested in (1) above), you have the information you need. Often in these problems, the information needed for the back-substitution is contained in the set-up. For the visual learner, you also then can show that the right triangle matches up with the algebra.

[@metalmanny0000]

The first time I saw this in class my head exploded. I watched the lecture again and it clicked. Thankfully we were allowed to use the formulas for the exam.

[@netflixyoutube3727]

7:50 "cluster-f*ck"

[@timjoyalle318]

I watched and benefited from many of your videos. A bit of a clusterfuck just made me smile. Nancy is real like me!

[@veronicadean4767]

Thanks! I'm in Calc 2 in College and your videos are awesome!

[@antoinecharlesdegaulle580]

same

[@revooluv]

nancypi iconic, legandary, savior of the universe

[@gp2580]

Thanks Nancy - I always found trig subs super difficult but you explained it really clearly. You've definitely filled in the gaps for me - I think I'll need a lot of practice but your video is amazing help! - thank you!

[@ompatel621]

did you end up learning it?

[@jkdog3242]

Nancy saying "this is a bit of a clusterfuck" is absolutely fantastic

[@loganwright296]

Wow Nancy, you have helped me more than any teacher I have had. Thank you for explaining calculus in such understandable terms. This is definitely what you were meant to do.

[@u235u235u235]

common to say that when trying 2nd, 3rd, or 4th, etc. to learn something. you're really just learning deeper based on info you already know. in other words, if her lesson was your first you'd still be in the same place your first teacher left you in.

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You are the best math teacher I have ever had. You explain things so people like me (the unwashed masses) can understand. In my opinion, you are the perfect person in every way. Thank you. Please keep teaching, I'm getting so much from this.

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like a math Megan Fox

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amen brother

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better

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minus the surgeries and strange thumbs

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Women do think...

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Except she can actually do what she does well lol

[@reecevdw]

I just want to say thank you. I am in university right now and have a graduate student that is teaching the class. He tries his best, but honestly, I do not learn anything from him. You have taught me all the class material.

[@silentintegrals9104]

Hopefully it works! Never give up!

[@rezazom27]

Great way of explaining that level of math. That is the way I used to teach my undergraduate students in different universit classes. The formulas are all known, you just have teach the students how to use them.

[@Twobarpsi]

She protec She attac She writes bac (wards)

[@padenzimmermann1892]

No cap

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She Just flipped the video to the side. She’s right handed.

[@Twobarpsi]

@@philnightjar1971 😃

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the joy i feel when i see your videos pop after searching a topic

[@ashleyjanelleyar]

Coincidentally, I have my exam tomorrow morning lol. You are very helpful, thank you so much Nancy!

[@Lup3r]

I can legit focus on this video really well. My prof has a greek accent so it's sometimes hard to hear and pay attention to what he's saying but this is really really great! Thanks for the upload Nancy!

[@Hitmandan]

Your videos, as always, end up being the best and help me out where I was struggling! Thank you so much for these videos!

[@keithperry6137]

Awesome tutorial. She just explains what it is. Where were videos like this when I was a math student? Why didn't my professors just explain it this simply?

[@Mark-dc1su]

Thank you, Nancy!!! I've been learning Calculus as a personal project and this video was fantastic!

[@kriselleannqueja7374]

The only video of you that I don't understand. Maybe because I'm sleepy or simply I don't understand at all.

[@billwindsor4224]

@NancyPi - this is excellent -- careful and thorough walk-through, and encouraging words sprinkled throughout. *Thank you!*

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It's crazy to think that someone could be so blessed to be the most intelligent person I've ever come across in math, is also the most beautiful person I've ever seen!

[@lucybusselle2182]

Bro this just explained trig sub to me so much better than my prof thank you so so much.

[@Valadez14]

Thank you explain so well. My Teacher tried to explain it. I was sooo lost. I understand it because of you. Thank you.

[@CapsCtrl]

she is all i need in my life

[@emmanuelnava6582]

I threw her a like the moment she said “clusterfuck”. Amazing

[@apmfer]

Time stamp im not watching this whole thing

[@emmanuelnava6582]

@@apmfer nobody caaaaresss

[@benisawesome868]

@@apmfer 7:50

[@apmfer]

@@benisawesome868 thx man

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LITERALLY UPLOADED JUST IN TIME FOR MY EXAM TY ILY

[@EduardoGarcia-hz6gu]

This really helped my out on my test. Thanks Megan Fox, very cool!

[@apmfer]

Hahahhahahahaha not even close

[@jayyyzeee6409]

@NancyPi, I like your sarcasm at 5:37. Thank you for these videos. I learned these things many years ago, so this is a refresher for me. I have no purpose for it other than pure interest. Yours are my favorite. I like the pacing and format. You're also extremely easy on the eyes ;)

[@SussyYT]

she dont want you bro

[@afifsiddique8856]

Not only your Maths but also your words even help me a lot! Thanks! ❤

[@bradley7118]

Got damn, there’s sooo many steps that you get lost along the way arrrrgghhh. I gotta practice some more because I felt like I got left behind since from the jump.

[@GabyP17]

Can you make more calculus videos 🙏🏼 past derivatives & implicit differentiation. Such as related rates? Or derivatives of inverse functions?

[@runtrunks2148]

Yes, please.

[@kumarupendra]

watch my video for all types of Differentiation Problems

[@navjotsingh2251]

kumar upendra it says your channel has no videos .......

[@rajendramisir3530]

Excellent explanation of this technique using good examples. Very helpful and insightful for me. Thanks Nancy!

[@alex-dn9to]

yay love when i get a new topic in class that nancypi goes over!!

[@kelvinndaki4543]

I love you Nancy. You are really my heroine in my time of need#calculus2

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Dont do drugs bro

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Haha

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This woman SAVED ME during calc2

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try fent

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Your teaching method is totally different and amazing. loved it dude❤️

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You are my hero. Love all your videos, but I probably need this one most!

[@dalekriens1397]

cluster what? that was to funny.. I think I learning something. thank you. you are a good teacher.

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I was just humming along until the "cluster f***" and that made me sit up and pay attention. Lol

[@SirWolf2018]

I wish I knew this channel during my university years. These vids highly motivate me to learn and get better at math!

[@rma1563]

I have been following the integration playlist and I was able to understand most of it very well thanks to her, but this sh!t went over my stupid brain. Goodluck folks.

[@yoafd]

same here bro😂 but we will figure it out 💪🏾

[@mrchapsnap]

When she says "don't be scared of the trig," it sounds like she's saying it to herself as much as she's saying it to us...

[@ce-qp2os]

I would never miss your class if you were my teacher😍

[@gabriellamonica-sleczkowsk9647]

Thank you so much! Went from not understanding at all to fully getting it bc of this

[@leeducationjames5558]

Literally the bhaddest. U the finest teacher I’ve ever seen bruh.

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Life would be over without nancy pi+

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A beautiful woman who describes a grouping of trig integrals as a clusterfuck has a very bright future in teaching! I, for one, will forever be your disciple.

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Will you ever do partial fractions?

[@SussyYT]

wipe your mouth when your done!!!

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my heart melted when you said “clusterfuck.”

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Nancy you make math so fun and exciting. I love it and your voice. Your a calculus princess.

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Less ego more soul

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yo nancy wheres nancy merch

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yea we need merch NOW

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I would cop ngl.

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Dude is it just me or is it impressive she can write in mirror reverse. Seriously though channels like yours make math seem simple for everyone :D

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You do realize people don't actually do that right?, Its pretty obvious considering you're smart enough to learn integration.

[@siangu6210]

Thank you so much. You are the best math teacher I found online.

[@ChuckyMC75]

At 2:12 that integral can be done by partial fractions (no need for inverse hyperbolic trig functions there).

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Your voice and energy are so damn comforting. Are you like 27 years old ? You have a master is mathematics ?

[@RohanSingh-sl4gh]

I'm not a maths student

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lol

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Then gtfo

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I know why you are here

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Me neither but PHYSICS :c

[@Twobarpsi]

Or English, obviously 😂😂😂

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whenever you say "Okay, let me help you out.", my spidey sense tells me I'm not going to understand what you are about to write

[@SolidSiren]

But she lays EVERYTHING out for you. She explains every single step of every single math concept. Every. little. detail.

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@@SolidSiren defs not everything. But to each their own favorite online math teacher

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“I know I love finding new integration techniques in my free time” giiiiirl in need that brain ♥️

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Thanks Nancy! Saved me right before finals :')

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😮 *OMG* ... Thank you, *Nancy Pi* (π) ... 👍 You're, *SUPER-AWESOME* ... 😍😍😍

[@machobunny1]

Nancy almost seems to have elastic arms. Look at how far that girl can reach. I think the writing backwards part must be a software trick, although, I had one professor who could write with both hands at the same time, both sides of the equations went up at the same time with the result that NO ONE IN THE CLASS could ever keep up. Thanks Nancy...You make it all much more interesting, and thank you for the candor...YES, we all use tables whenever we can. I've lugged Abramowitz and Stegun into more than one open book final exam.

[@SussyYT]

touch grass

[@samyud1819]

AHHH this is so much simpler an explanation ! Thank you I wish it was taught to me like this

[@abdullatif9108]

Nancy you made calculus 1 easy for me... now i am taking cal 2.. and i have no clue how to do it without you... please make it soon

[@vjzb3]

I love math, and I’ll have all of this down pretty soon. But at first exposure... it’s kind of insane how wild solving these problems is getting lol

[@Prxwler]

I feel literally the same. I'll be probably learning all of this stuff this year, and it amazes me equally as much as it scares me.

[@vjzb3]

@@Prxwler as long as you don’t quit, and just keep doing your best, eventually this stuff will feel like second nature to you. Be patient with yourself, you can do it 🙌🏼

[@Prxwler]

​@@vjzb3 Thanks man! I hope you can achieve whatever you're seeking too. Also, giving it another look I see that your original comment was posted 1 year ago. Have you already gone through this? If so, how was it?

[@vjzb3]

@@Prxwler I went as far as I needed to to get through my active math course at that time. Have taken the last year off from University to start full-time work as a software engineer, but will be revisiting this material once I return to school next semester. I can tell you that these later exposures to material like this are less daunting than they were previously. A huge part of learning is all about exposure. The more times you see something, the more times you think about it, the more times you practices it, the more familiar it becomes. I’m looking forward to my next math course and getting more exposure with these trig substitutions and beyond!

[@Prxwler]

​@@vjzb3 Woah that's really cool mate, and it seems that having your goals clear definetely helps you. Best wishes for your next math course too!

[@JagooTheBoss_24-fj]

you could pass easily if you watch her videos the night before the exam.hats off mam❤️

[@krutharthramegowdayd5291]

Instead of 5x= 2 tany, u can also take x^2 common ouutta numerator and take 25+4/x^2 as u and u can easily manipulate to take rest of the integrand as du and life is good

[@viditbatta2434]

I found it really interesting to hear her say clusterfuck in the nicest way possible. had to rewind to make sure i heard it right lmaoo

[@ngym7]

Wow. Ur my idol. You just saved me for my midterm. Greatly appreciate it.

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Her really? I love her but like there are people like meryl streep and stuff in this worlds so 🤷 I mean shes not idol level

[@SussyYT]

Stop the glazing buddy

[@goodguy2627]

Nice job Nancy Thanks a lot for your explanations. And BTW super nice optical trick making it look like backwards left hand writing behind the glass. Awesome 🥰

[@aobakwekeatlhomile8226]

I love how she's not making it seem like the easiest thing ever

[@geekedslotter7948]

This is the most random recommend video I've ever gotten. I haven't had to do this since freshman year and I don't plan on doing it this year or my senior year.

[@rakshitbhoj9987]

was searching for this top like hell thnx alot ma'am

[@Raining345]

Thank god you wrote the rules, I was not prepared to research those and my professor didn't show my class like 50% of them -_-

[@deephish]

Hey Nancy, You made a mistake. at 14:37, your final answer has an x/2 where an x/6 should be. I think, that is if sin(theta)=x/6.

[@stgeorge98]

I thought the same thing. Wouldn't the sqrt(36-x^2) also be /6 if cos(theta) = sqrt(36-x^2)/6.

[@emanuelgonzalez7213]

@@stgeorge98 36/18 =2

[@drankankay3185]

Your teaching style is outstanding !!!

[@thorodinson7467]

Thank you Nancy, Very Pi!

[@thewyk]

Ty Kanye

[@jayrum7303]

I'm in my forties revisiting this stuff, but when teacher dropped the f bomb I giggled a bit like a middle schooler would have. Anyhoo, it was nice having a laugh during a total clusterfuck of a subject!

[@ethenjoes2421]

Would u mind clearing my doubt:- My question is at the time stance 24:30,how did 64|tan|cube turn into tan square theta??

[@anthoniflores203]

This link: prnt.sc/th68iu you'll find answer to your question friend.

[@WarriorBane]

Did u find the answer?

[@ethenjoes2421]

@@WarriorBane yup

[@ethenjoes2421]

@@anthoniflores203 thanks for the answer

[@noneatthistime]

I missed you Nancy! I'm taking pre calc for summer. keep me interested now :)

[@_hydrogelic]

This thing is the final boss of integration

[@KewlKid]

Very helpful for my Further maths a level

[@ezequieltorreslagunas6378]

The steps you use are ordered and with a lovely touch

[@rhaegarbaguhin]

best math tutor online. we all know why.

[@whoiswho-e6h]

Your teaching is very good. And also the course is well. Excellent.

[@juliomiranda2131]

I took cal 2 about 4 years ago and man I remember how much I hated trig sub!! took like an entire page landscape to do 1 problem