t-Formulae ~ 03 ~ Used to Evaluate Certain Integrals

One of my subscribers asked me to explain what t-formulae are, and how they are used in trigonometry. This is the last of three videos that I created in response to that request (thank you, Kristyn).

Because t-formulae allow us to convert trigonometric functions into fractions involving polynomials, we can use them to change certain trigonometric integrals to 'integrals of fractions containing polynomials.' This can often (but not always) be an advantage.

Such substitutions are particularly valuable when the original integral only involves single instances of trigonometric functions and no powers. In fact, this is when t-formulae are almost exclusively used. In other words, we look for integrals that involve uses of sin(x), cos(x), and tan(x), but not powers of these functions.

The fraction that results will contain two polynomials. These may produce a logarithmic structure, or may require a division to take place, or may lead to the use of partial fractions, or may even produce inverse tangent functions, for example ... but many of these structures are almost instantly recognisable and experienced mathematicians know how to 'handle' them.

Because the complete solution of this integral has been accomplished in another video, I only spend sufficient time with the integral to show how the use of t-formulae allows us to convert it to a much more useable and recognisable form.

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Best wishes for your study and your mathematics!

[MrVoayer]

Great example that clearly shows how helpful t-Formulae could be ! In this particular example there was a small nuisance od finding dx in order to use t-Formulae substitution, but after that everything develops quite nicely. Thank you, Graeme, for a valuable demonstration!

[AkashSharma-nmjc]

Hi Graeme!
How are you nowadays ?
Happy new year 2021 !
Still remember how inspiring were (still are !) your "How to Study" series.
I watched them in my middle school and still is helping me when I have passed through my high school and my university (almost).
Thank you so much Graeme!

[carvelbell]

Your tutorials are excellent. I love your channels, excellent teaching techniqiues, it help me to see and learn concept clearly.

[nativeronica]

Whenever I am not sure about something in maths, I head over to your channel 😊

[elysiangrowth]

Great to see how many comments there are here recently paying homage Graeme!

I'm Harry, the young lad who illegally drove a car to one of your maths lessons when he was 15 because he was late.

Doing nothing related to maths today 10yrs later, but I remember on at least a weekly basis what you told me about persistence and work ethic, like your one-armed pullups

Thanks a load Graeme, hope you're very very well!
HB

[arkanmanva]

I came across our channel while looking to learn some concepts and you were the only one that had had it. Your content is so helpful.
Also Sir when is your next video coming out? Hope you in good health? I'm really anxious as you haven't posted in a long time (minimum 1 year), but you have replied to many comments. Please do start posting again.

[AbhinavKrishna]

Hi Graeme,

I was fiddling randomly on the internet and found your videos. I must tell you that I've become a fan of yours. I hope to see your next upload soon.

Regards,
Abhinav

[SPELLTHEWORDCAT]

Hello sir how are you doing this is vikhyat here you were the only one who cleared my doubts of integrals i am now in a college doing well and how about you i hope you are too, and i hope you will be like this

[coffeebeans]

thank you so much for these videos you explain everything perfectly it's so incredibly helpful :)

[abrarmansur]

Thank you for the videos sir. May I ask how you are doing these days?

[lazyman]

Hi, i hope you are reading this, i want to tank you forward for all the immense efforts you have done to make maths more understandable, i wish u the best,
Second, i want your help and opinion, i have my baccalauréat math exam tommorow, i had i would say inexperinced mistakes i have done through the year mainly i would say : that i start doubting my informations the day if the exams, i start noticing new things, careless mistakes, and a lack of concentration, if you have any solutions i can do regarding this i hope you provide, thank u !

[coffeebeans]

Also wishing you all the best and really hope you are doing well

[Socrates...]

Happy 2020, I hope you are able to make more content this year.

[sahilsharma]

Hii sir
How are you sir?
Where have you gone ?
You didn't post any videos from so long
I thought you might be covering chapters like functions or limits or more calculus. You're an amazing professor and I would love to see more of your content

[Socrates...]

I hope you come back. Your contribution is special.

[ifmohammed]

Hi teacher really missed You and Your videos come back soon

[AleksandarKospenda]

Are you alright bud? You plan on making any more vidoes?

[Scerttle]

Not entirely sure I understand what's going on in this video, mostly because I'm not at this level. But I've bookmarked for when I am so I Can come back :D

[akifismail]

why are you not making videos anymore :(

[speedcubesolver]

Hi!
I hope you are good and will come back :)