I want all trig functions in one integral!

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Learn more calculus on Brilliant: 👉brilliant.org/blackpenredpen/ (now with a 30-day free trial plus 20% off with this link!)
how calculus teachers create integrals for your test! I took me about an hour to craft this integral which features all the trigonometric functions, the natural logarithm function, the pi-th power, and a derivative for the denominator. This integral looks scary but the solution is actually very straightforward! Enjoy! #calculus #math #trigonometry #blackpenredpen
Check out my 100 integrals for more integration practice for your Calculus 1 or Calculus 2 class. • 100 integrals (world r...
0:00 how calculus teachers create integrals for your test
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Thank you all!

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28 июн 2024

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Комментарии : 570

@blackpenredpen 8 месяцев назад

Learn more calculus on Brilliant: 👉brilliant.org/blackpenredpen/ (now with a 30-day free trial plus 20% off with this link!)

@pjshorts4129 8 месяцев назад

Thank you for this information but I really want you to tell some basic for integration thanks you

@jonathanv.hoffmann3089 6 месяцев назад

Thanks from Brazil!!! 🙏♾🎁👏🏻👏🏻👏🏻🎉🎊

@mohamedsamsudeen7694 4 месяца назад

@blackpenredpen now solve Integral {[(e power sinx + sqrt(cosx+tanx)] power pi} ---------------------------------------- [ln (ln (cosec x)) + sqrt(sec²x+cot²x)]

@dreamdefence7952 4 месяца назад

Nice what is your country name

@dreamdefence7952 4 месяца назад

India

@JCCyC 8 месяцев назад

What the student really needs to learn here is not Calculus, but to control their own panic.

@CliffSedge-nu5fv 7 месяцев назад

Exactly! It is natural to feel intimidated and want to reach for the biggest weapon, but this beast can be taken down barehanded.

@simonmiller1361 4 месяца назад

Biggest thing I realized in calculus is it's not that bad if you master trigonometry

@itsblu4735 3 месяца назад

Exactly! Like at home sure i have all the time but at the exam I'd feel the cold sweat dripping down my face fr 💀

@kalculate1297 8 месяцев назад

i did not expect such a clean solution for this integral

@user-qz6sh7dy2h 8 месяцев назад

It was way too easy question though

@canyoupoop 8 месяцев назад

@@user-qz6sh7dy2hayyy same pfp

@UKYusei 8 месяцев назад

@@canyoupoopwot da fok

@goated9415 8 месяцев назад

@@canyoupoop YO LUFFY wyd learning calc bro😭😭😭

@mudit2960 6 месяцев назад

thats the beauty of math!

@fortpile 8 месяцев назад

As a high school cal student, this is, so far, the most satisfying integral I've ever witnessed.

@pitoachumi2663 7 месяцев назад

really? come to india baby ull find satisfying triggers everyday then

@fortpile 7 месяцев назад

@@pitoachumi2663 oooo

@user-qg4el8lq9h 4 месяца назад

@@pitoachumi2663or just use the internet

@seemaverma7671 3 месяца назад

You better work in your tech support @@pitoachumi2663

@dhanyajishin8731 Месяц назад

@@pitoachumi2663 bro u r so right. I just started calculus and it is already hard. But the truth is this one was extremeley satisfying. But the truth is that in India the most difficult questions would be very easily solved by the teachers but we end up gettting dumbfounded by the same questions. It's annoying but fun.

@Wmann 8 месяцев назад

“Everybody’s here” I love that part.

@ozzy_animations17 Месяц назад

smash bros but it’s an integral

@General12th 8 месяцев назад

What a glorious result! I'm going to make this extra credit the first time I teach a calculus class!

@blackpenredpen 8 месяцев назад

😆

@itachu. 8 месяцев назад

Epik

@caroot1085 4 месяца назад

All your students who watch blackpenredpen: STONKS!

@EvilRamin 18 дней назад

@@caroot1085 If they watch this, they deserve the score

@MichaelZankel 8 месяцев назад

It’s always so funny to me when you add the sin(x)^2 + cos(x)^2 just to make it look harder but it’s just 1 😂 I love it lol

@DiverseDose11 8 месяцев назад

If you love physics solve The speed of a motor engine decreases from 900 rev/min. to 600rev / min in in 10 seconds. Calculate: The angular acceleration Number of revolutions made by the motor during this interval How many additional seconds are required for motor to come to rest in the same rate

@DiverseDose11 8 месяцев назад

If you love physics solve find the value of ratio of specific heat capacity for the mixture of gas containing one mole of nitrogen and 2 mole of argon if Gamma of nitrogen is equal to 1.40 and gamma of argon is equal to 1.54

@DiverseDose11 8 месяцев назад

If you love physics solve 3 moles of an ideal gas undergoes a reversible isothermal compression at 27 degree Celsius during this compression 1850 joule of work is done on the gas what is the change in entropy

@NiceLol-dl6lq 8 месяцев назад

@@DiverseDose11i was literally revising this exact topic ill solve it when i finish revising

@DiverseDose11 8 месяцев назад

@@NiceLol-dl6lq yeah it will be nice 👍

@gamerpedia1535 8 месяцев назад

Time to try it out! sin²+cos²+tan²=1+tan²=sec² (1-2ln(cos))^π ln(csc)+ln(sec)+ln(cot) = -ln(sin)-ln(cos)+ln(cos)-ln(sin) = -ln(sin) d/dx(-2ln(sin)) = -2 × cos/sin = -2cot So the integral becomes sin×(1-2ln(cos))^π / -2cos u = -ln(cos) du = sin/cos dx So (2u+1)^π du Where u = -ln(cos x) Now another substitution w = 2u+1 dw = 2du So -¼×w^π dw Integrates to w^π+1 / (-4)(π+1) w = 2u+1 u = -ln(cos x) w = 1-2ln(cos x) (1-2ln(cos x))^(π+1) / -4(π+1) + C

@R8Spike 8 месяцев назад

Lmao it has the translate to english button

@mandarbamane4268 8 месяцев назад

@@R8Spike and I got Hindi button & sine = sin (pronounced seen) lol

@lolynoras-ss8qs 3 месяца назад

why didn't you take the w as 1-2ln(cosx) from the beginning it's easier than two substitutions

@TRT_MOOSIC 5 дней назад

@@mandarbamane4268 for me it was sine = syn

@johncraig9310 8 месяцев назад

This is the only guy that can make math fun. Keep up the amazing content!

@blackpenredpen 8 месяцев назад

Thanks!!

@DownDance 8 месяцев назад

5:12 My favorite scene because we all know what he means 😂 This problem was actually quite easy for looking so difficult. The only things I didn't instantly see is the u-sub, because I didn't instantly know the derivative of sec(x) Great video!

@CorrectHorseBatteryStaple472 8 месяцев назад

Stuff like that is why I keep watching this guy

@user-hh4nh2zj1t 2 месяца назад

idk what he means

@herobrine1847 2 месяца назад

What does he mean

@MathZeimer 8 месяцев назад

There is nothing more satisfying than terms that perfectly cancel/match out, leaving a neat answer!

@JoseIgnacioZapata 4 месяца назад

You are so calm, smart and good natured I can't help but smile as I watch you solve these. Good work!

@nvapisces7011 8 месяцев назад

When i saw this, it sort of look solvable and now i see this being solved makes it so satisfying

@tomasthemas 8 месяцев назад

I love how real the problem creation is. Start with a random smattering of operations and functions that look scary then adjust it until a solvable problem appears.

@nomad-1776 21 день назад

I'm just impressed he managed to fit it all on the whiteboard.

@frogmouth1416 7 месяцев назад

I’m a high school student rn and I’ve been watching blackpenredpen for a few years now. Well I’ve finally made it to calculus in school and this is the first time in my life I’ve actually understood what happened in the vid (normally id just watch and be interested), pretty cool!

@ricardomorales213 8 месяцев назад

please make more "insane" looking integrals that are actually fairly easy! I loved the way you break down the solution for something so ridiculous looking :)

@sonictheone4568 4 месяца назад

This was a fun one to do! Cool how all the identities and the u-sub lines up nicely

@Zhinoi 8 месяцев назад

This is one of the most beautiful integrals I've ever seen...

@Dantido 7 месяцев назад

That was just beautiful. I've never seen such a complicated integral become so easy in such a low amount of time.

@qu3nt0r 7 месяцев назад

As a student that has no idea what half of those term means, it was enjoyable. I’ll take a look at this again once I learned those terms.

@UtpalAman-tm1tf 3 месяца назад

Didn't expect such a easy solution

@mineapple6726 8 месяцев назад

It’s always good when teacher give you things in a test you’ve never seen before, like cscx, secx and cotx

@hyacinth5040 8 месяцев назад

True, I love when they teach you the absolute basics and then immediately give you a classwork on the toughest parts without the chance to even study 😀

@wistfulgraph Месяц назад

it feels so blessed to understand this after taking calc bc

@brunomcleod 8 месяцев назад

I have not got the faintest idea what he just said, but the way he explained everything made me feel like I did

@bylightIta 8 месяцев назад

I'm in high school and this year we finally did the indefinite integrals. Now I can comprehend some of this man’s videos. Nice solution

@Kimeters 7 месяцев назад

love this one, thank you

@gmjackson1456 8 месяцев назад

Nicely solved!

@Rishith198 4 месяца назад

This was a very interesting one to solve Loved it! :D

@infinitexgg Месяц назад

I got everything! nice job

@sk3105 8 месяцев назад

beautifully explained

@omurice8534 3 месяца назад

NO way how did it fit so perfectly

@MrWaltjam 3 месяца назад

as a 50 year old I recently(2020-2022) went back and got my associates degrees after a debilitating health incident that left me reconstructing my base of knowledge. I also wanted to work my brain and ended up taking physics and calculus 1 and 2. At the community college level. I still have memory issues but these videos keep me atleast familiar with the concept of integration. I did well almost coming out with an A but for fractions of points in both.. graduated with a 3.92 gpa. the final Bs in calc I and II got my 4.0.. I can now help my highschool and middleschool children when they get stumped in their advanced math classes. I love the depth to which you dive in explaining your process and the sloutions..Your humor is also not lost on me. Fun makes learning more bearable in difficult situations. I only wish my professor was so concerned with teaching the process. After the few example in the book, she assigned out homework. She wanted it the way she taught it but left us to "figure it out" no pun intended.. I would have drowned but for math is power, professor leanard and some of the others. I realized the value of not only knowing but proving the trig identites alittle too late but now I know. I may go back and finish the bachelors in meterology but am leaning toward mathematics just for fun. Keep up the good work, you are opening doors in the minds of many who were traumatized by higher maths as evidenced by the comments.. my only question is whether all the variations of blackpenredpen is actually you? there are several, its hard to tell these days.

@Larsbutb4d 23 дня назад

Watching this a month later, where I learnt a bit of diff and int calc, feels so nice to me bc I now understand what these are.

@aquss33 5 месяцев назад

This is possibly the greatest integral I've ever seen!

@user-zk1dl2fv7y 8 месяцев назад

Where is the hyperbolic and the inverse of all the functions (sin, cos, tan, csc, sec, cot, arcsin, arccos, arctan, arccsc, arcsec, arccot, sinh, cosh, tanh, csch, sech, coth, arcsinh, arccosh, arctanh, arccsch, arcsech, and arccoth). Then you will have every trigonometric function in one integral. Here is one: Take the indefinite integral (Using the complex definitions of the trigonometric functions to get values for all x) of: sin(x)^(tan(x)) - cos(x)+csc(x)sec(x) - arcsin(cot(x))/(arccos(arctan(x)) -sqrt(sech/arccoth) + (arccsc(ln(x)))/(sinh(cosh(arccot(tanh(x)-arcsec(x)))))+csch(π*x^(arcsech(x)-coth(x)) + arcsinh(x) - W(arccosh(arctanh(sqrt(x)))) + arccsch(x) + arcsech(x) + arccoth(x) 🤣

@advayamshud484 8 месяцев назад

BlackpenRedpen I really am your fan and i love your videos, your way of explanation fascinates me and I understood every topic of your video, even my mother encourages your teaching and I had watched almost all your videos and I still watch them! Thank you for your teaching and making math enjoyable I had learnt many things from you, and I encourage you to keep going! Al the best!!

@blackpenredpen 8 месяцев назад

Awesome, thank you!!!

@gocomputing8529 8 месяцев назад

Great video! The trick here was to keep the sin in the denominator and the sec in the numerator by playing with the negative exponent

@y_auc60seb20 8 месяцев назад

This was a beautiful integral

@brinoinfantprabu6197 8 месяцев назад

Your alway best. It seems to be very hard but after your explanation it looks like very easy one. Keep rocking on mathematicians

@blackpenredpen 8 месяцев назад

Thanks a lot 😊

@samhess78 23 дня назад

Best math teacher. You are incredible

@omkaslikar912 27 дней назад

We completed indefinite integration in our highschool yesterday and first thing in my mind was to visit bprp... And I did solve it on my own before watching his solution. Yay!

@mcgaming1172 8 месяцев назад

As a 8th grader I understand absolutely nothing but I like the way he is teaching

@user-oq7cx2rb4t 8 месяцев назад

do you not do calc in year 8 in america?

@mcgaming1172 8 месяцев назад

@@user-oq7cx2rb4t I am not in America, in our country we learn calculus in grade 12 and college

@loverofstyanax 8 месяцев назад

Bro where do you live that you are doing Calc in year 8?

@antiarezzo7630 3 месяца назад

Instantly assuming he's from the us@@user-oq7cx2rb4t

@marcorizzo3854 2 месяца назад

@@_.Max.i.mus._year 8 is like 14 right? Too old for calc 1. Im my country we have math phd by age 16

@Peter_1986 7 месяцев назад

This is the kind of stuff that you throw into Wolfram Alpha when the teachers no longer care about the exact solution steps for integrals. That was a really pleasant discovery for me when I started the math courses after calculus, when the teachers started being like "just find the numerical solution in some way".

@shipsteer2891 8 месяцев назад

This looks so hard but is so satisfying

@jordierinsyaputra1932 6 месяцев назад

This is amazing

@KellyBeeSan 8 месяцев назад

I was so satisfied when I followed along and got the same answer ☺️

@EduardoViruenaSilva 8 месяцев назад

Amazing!ly simple!!

@vasco35 23 дня назад

It turned out really easy

@EC4U2C_Studioz 7 месяцев назад

Try doing any integral to the e power.

@schroodinger 5 месяцев назад

Waiting for hyperbolical version of this integral :)

@revtheobbyist2222 3 месяца назад

wow its so much easier than i thought

@a_man80 5 месяцев назад

5:46 I thought he is going to say "pi is just 3"

@embersofthesunset 8 месяцев назад

this solution was really satisfying, haha!

@krypt.shaala 21 день назад

I love how there are black and red whiteboard pen boxes in bulk under his table 🥰

@peterchege4616 8 месяцев назад

Can you explain the vibrating string string 2nd order PDE in a video

@ANTI_UTTP_FOR_REAL 8 месяцев назад

This came out on my birthday and I didnt even notice it 😂

@acloaker600 2 месяца назад

I love it how brother gets all gitty to do this problem and then starts cranking 90s with the dry-erase marker.

@iamboomtuber7467 8 месяцев назад

that double tap of pen made me scared, but i mustered courage to face it

@instinx9154 8 месяцев назад

This was such a satisfying one to do on my own. The way everything cancelled as I used a couple trig identities and a u sub was so satisfying

@chills_tiny_mom 8 месяцев назад

Man I love trig identities 😊

@hitmanjack2495 8 месяцев назад

Hey I was just wondering if u could make a video on proof of chain rule for differentiation.

@Yossus 8 месяцев назад

I love these videos but I have absolutely no idea whether there's a system behind the pen colours or if it's just a semi-random choice thing.

@blackpenredpen 8 месяцев назад

It’s systematic. The red is for important parts, new parts or side notes.

@beyzagogebakannn 8 месяцев назад

Great video! Like a puzzle. It’s develops trigonometry and integral.

@horny_emoji 8 месяцев назад

As a comp. Aspirant , this one felt good

@peterchege4616 8 месяцев назад

Can you do content on partial differential equations (PDEs)

@mattreichmann8118 7 месяцев назад

They're actually super simple! If you're trying to find the derivative of x^2+y^3+z^4 Partial derivative of x -- simply treat Y and Z as constants, so you get 2x. That's it!

@lazarmendel3784 4 месяца назад

@@mattreichmann8118 that's not what a pde is

@xinpingdonohoe3978 4 месяца назад

It's not easy. They're very abstract, so methods work for them even less than for regular ODEs. I guess one trick is assuming separability, and seeing what transpires. So you have a function f(x,y,z). Maybe you're solving something like f_xx=f_yz. Something that looks separable. Just assume f(x,y,z)=X(x)Y(y)Z(z) and see if you can derive a set of solutions.

@divedownl5767 8 месяцев назад

beautiful problem

@shres2712 7 месяцев назад

Hyperbolic trig next?

@DevRaj-my2wo 8 месяцев назад

These are the type of questions which should be left without a thought

@keirgordon8646 8 месяцев назад

I don't understand any of these but I still watch ur vids

@BusyBlueLion 8 месяцев назад

Thank you

@proffessor3064 8 месяцев назад

Those expression after taking the substitution 😂♥️

@nocturnalvisionmusic 7 месяцев назад

Finally got back to watching you again ^^ Love that suit profile thumbnail 😎

@jamesd9567 8 месяцев назад

This problem rules, and I love that I'm far enough into calculus to be able to not only follow along, but (possibly) solve this type of integral. The only 2 parts I was worried about were taking the derivative of the denominator while leaving it as 2ln|csc|, but it was simple to still get -2cotx. The other part that I'm not sure if I could find is what to set as u, my professor has been very generous with what u is going to equal, but looking back at this problem I would likely originally try setting just secx as u, finding that doesn't work, and then hopefully try setting everything within the parentheses, but we'll see. What strategy would you recommend when trying to find u? I have my exam over this section Tuesday, wish me luck!

@jamesd9567 8 месяцев назад

Looking back at this with some actual sleep... All this is is a u-sub (and a lot of simplifying) lol. All the trig threw me off since we're doing trig sub and fraction decomp. right now. It's insane that cal 1 could do this, I'm definitely sharing this with my previous professor

@user-pv5hd1vu1t 8 месяцев назад

also him not putting absolute values when doing ln(csc^2(x)) = 2ln|csc(x)| = 2ln(csc(x)) is because the integrand restricts csc(x) > 0 by nature similarly with ln(sec^2(x))

@eshwarthammineni7911 7 месяцев назад

Coming from a high school student who just learned integration, my teacher recommended I pick an inside function as u, and if the derivative is seen in the expression in the integral, that most likely is u. Again, I might be completely wrong.

@user-pv5hd1vu1t 7 месяцев назад

@@eshwarthammineni7911 this generally is a good tactic to try for some integrals.

@selvinmoh9402 8 месяцев назад

cscx = 1/Sinx

@lukaskamin755 7 месяцев назад

I always feel uncomfortable when even power (2 in this case) is factored in front of ln, while since then the contents of the ln may turn (and we know for sure that it does so in case of trigonometric functions), so strictly speaking you have to put a modulus (abs. value), so that ln (y^2) = 2 ln |y|. But I constatntly see this thing like many alike are neglected while taking integrals, I wonder does it always works? If yes, why, If no, when it works?

@ChadTanker 8 месяцев назад

i gotta remember trig identities

@technogamer1021 3 месяца назад

Actually got that right

@GreenMeansGOF 8 месяцев назад

This only works if you are in the first quadrant. Otherwise, you need to be careful about pulling the 2’s out of natural log.

@user-pv5hd1vu1t 8 месяцев назад

but in the original integrand, we have in the denominator ln(csc(x)) ln(sec(x)) etc which means it's restricted to first quadrant by nature

@ArchIVEDCinema 7 месяцев назад

Watching this made me realize how much being less than 2 years out of college made me forget how to recognize simple trig IDs, log rules and etc. I probably would've gotten this pretty quickly a few years ago 😫