the ultimate integral starter (u sub, IBP, trig sub, partial fractions & more)

[@blackpenredpen]

Time stamps I. Know your derivatives 1:06 II. Reverse Power Rule 8:54 III. U Sub 18:30 IV. Know the Famous Ones (part1. the famous first step) 44:00 V. Say NO to Integral Addictions 1:07:24 VI. Know the Famous Ones (part2. famous non-elementary integrals) 1:24:24 VII. Integration by Parts u-dv setup 1:46:26 DI set up 2:14:15 VIII. Use Trig Identities 2:50:35 IX. Trig Sub 3:32:00 X. Partial Fractions Decomposition (all cases included) 4:26:50 The file: docs.wixstatic.com/ugd/287ba5_9a2f59e31ee3469cbf1236ac69192abc.pdf If you enjoy my videos, then you can click here to subscribe kzbin.info Story about Anna. One day in class, I asked my students to guess my first job. Among all the random answers, Anna answered "McDonald's"! However, a guy next to her said "OUCH". Then Anna said "I know because I watched his videos!". This is why I wore my McDonald's uniform in this video. Story about Matthew. One day after my algebra class. A guy (yes, Matthew) had been standing outside waiting for me to finish my class but I did not know who he was. Then he told me that he had been watching my videos and could not believe that he saw me teaching in that classroom. He waited for me and we took a selfie. The following semester he signed up to audit my calc 2 class.

[@abhishektyagi4428]

SIR THE RESOURCES AND LINKS TO LEARN MATHEMATICS THAT YOU SAID IN YOUR VIDEO WITH fematika ARE STILL NOT UPLOADED IN THE DESCRIPTION OF THE VIDEO , please do upload those links

[@kamalrastogi5041]

Solve iit jee advance maths problems challenge

[@cm221b]

@@kamalrastogi5041 exactly

[@mjshaheed]

Hi, Please enable automatic closed captions for non-native english speakers like me

[@andrewnikitin7651]

can you show how solve double integral on area x^4+y^4

[@HDitzzDH]

He was at school from 5 pm to 10 pm just doing a bunch of integrals in front of a camera, that's dedication.

[@rashwienmohabier9376]

He was there till 11...

[@kamigod4181]

We need professor like that in engineering classes

[@好吧-h6k]

@@kamigod4181 😂😂😂😂😂

[@Jacobpppp]

Man is a legend

[@eddie-hm3rx]

+ holding a mic the whole time

[@ericabaez3033]

It so important to show a friendly and welcoming face when teaching something to someone, especially, something that by itself is so overwhelming like Calculus. Thank you for existing and being so good at teaching

[@blackpenredpen]

I am very to hear this comment. Thanks. Good luck for you on your calculus. If you need further help, please visit my new “just calculus” channel kzbin.info/www/bejne/m5CWq4l5lraWb7M

[@biggusdickus6436]

2 years late and now it's my turn to learn from your videos, i genuinely can't thank you enough sir You sound incredibly enthusiastic you made me​ a bit excited learning about calc lmao. Honestly though thanks, you're a great teacher and person :)@@blackpenredpen

[@tycobrahe7663]

“I didn’t go to Disneyland, I didn’t go to Disney World. I went to the U world!” Lol Love your videos. Passion is unmistakable. Keep it up!

[@blackpenredpen]

Thank you!!!!!

[@lambingsilog]

don't forget the theta world!!!

[@xd_ryuu3041]

Yea I love his analogy of the U world. Helps a lot :)

[@drstoned8523]

la pasion

[@ayushkrishn8048]

I never used to enjoy doing maths since middle school, but I have been wanting to self study maths for a long time using online resources without a fixed schedule and framework. Your videos really spark my interest in maths. Thank you for your hardwork and dedication.

[@user-fh2rw9zh8g]

Same! I wish i can send him a gift as you did😅

[@postholedigger8726]

Professor Chow has done a great job organizing this stuff. Due to his excellent lectures and the miracle of pause, rewind and review on KZbin, you can learn more in one week than you can learn in a year in a traditional classroom. SEAN

[@blackpenredpen]

Thank you Sean!

[@claudemunyakazi5417]

for sure🙏

[@G21lva]

When I graduate, I will come back to this video and thank you for contributing to another engineer in this world

[@stewartmoore5158]

Blackpenredpen: **makes 6 hour straight lecture** Morpheus: "He is the one."

[@blackpenredpen]

Lollll love the matrix reference!!

[@nathanielgolden3141]

without ANY irony or exaggeration, you are making the world a better place through these videos. Thank you so much.

[@kinglandon5831]

Not only did your videos help me find my passion for math, but you do them in a fun and charismatic wat

[@blackpenredpen]

Thank you. I am happy to hear it.

[@braeling3677]

If you’re currently in calc 2 this video is basically your life for the first half of the semester. If you can go through all these you’ll pass for sure.

[@term1234]

When u r in 10 th grade

[@Deebs-lx2fm]

I agree. He is a life savior

[@jony3781]

Thats litteraly your final exam in some german schools

[@hydropage2855]

What if I solved them all and haven't taken any calculus?

[@JirivandenAssem]

@@hydropage2855 what im wondering as well. Maybe when we get some calculus integrals on the test which we can calculate using our calculator I will just use these methods. And that would be funny. If u can do this u wil have a big advantage over other ppl if u study stuff containing integrals

[@alstamm1508]

Absolute lifesaver. None of the integration techniques are super hard to quiz on individually, but for my midterm I have to look at many integrals and know what technique[s] to apply. Very helpful to have a quick but comprehensive overview of them all in one place

[@blackpenredpen]

Glad to hear 😃

[@hanma1705]

This guy deserve's a Reward, he spends 1/3rd if his day in creating math videos. You sir, are a legend! I still can't forget how your videos helped me in my school exams, even though I scored less then expected, but I really enjoyed your lessons.❤❤❤

[@blackpenredpen]

Awww thank you Kumar.

[@hanma1705]

@@blackpenredpen I pray to God, you get whatever you want in life. Just keeping blessing us with these videos.

[@hamzanazim5943]

1/4*

[@LewNeko]

@@hamzanazim5943 24h - (6 to 8hours sleep) = 16 to 18 hours active. 6/16 or 6/18 is approximately a third of his day or even more.

[@hamzanazim5943]

@@LewNeko i was assuming it to be 24 hours, doesn't matter the guy works really hard he is in love with maths in the end that's what matters i enjoyed doing the integrals with him 😁😁

[@loopje]

This is false advertising. I came here for black pens and red pens, but he started using green pens too

[@blackpenredpen]

loopje hahahaha

[@stpuwl468]

4:33:15 it was for Christmas.

[@SyberMath]

I like orange pens! :)

[@joeljose182]

Oh yeah serious crime

[@aashsyed1277]

@@SyberMath HEY!!!!!!!!!!!!!!!!!!!!!!!!!! SYBER

[@maheshpadmanabh6564]

As an experienced professional in engineering field, this is the best revision for the basics.Thanks!

[@blackpenredpen]

Thank you!!!!

[@nasimuddin813]

How

[@alihansencerberk9670]

I'm currently studying for the physics olympiad. My school doesn't have a good olympiad teaching system so I have to learn most of the subjects by myself. And I couldn't understand integration. Thanks to this video, a lot of my questions are cleared and I can integrate much easier. Thank you very much for making my life and olympiad journey a lot easier!

[@zwatotem]

Yes man, you're right: "Usually the hardest step is the hardest part" (44:44)

[@blackpenredpen]

ɯǝʇoʇɐʍZ Lollllllllllllllll

[@jitulsarma1388]

I was searching for this comment 😂

[@pixelatedparcel]

Semi-retired 62 years old, learning math for fun. Recently discovered your outstanding channel. Watched all of this wonderful 6h video: not a dull or confusing moment. A real privilege to have you as a teacher. Congratulations and sincere thanks!

[@eathanehanthnaidu11a40]

The sweetest, the smartest, and the non grumpy math teacher. How is it even possible!?😆

[@blackpenredpen]

😆

[@AKSHATAHUJA]

Doesn't matter from where you had worked before. Every thing is about knowledge

[@blackpenredpen]

Yup

[@avimaybe]

@Sebastian Le Roux food 😍

[@DrunkMagnus]

And the ability to transfer the knowledge Which bprp does flawlessly

[@swiftyt6558]

i am proud to say i finished this video in 2 sittings, i have no idea how you could stand through all of that, that is massive dedication, i attempted 90% of the problems on my own, and almost always got those problems correct after you had explained the method for the starting problem, i got all 3 partial fractions right at the end on my first try. all in all, very epic, now i will no longer fail my test in 2 weeks

[@GoodSmile3]

Ok basically whole semester in one video. Would be so useful a couple of years ago for me. At least now students have an ultimate guide. Huge respect man.

[@inmer3216]

The fact that you were able to provide us with such tough material in such a fun and very well put together video is quite amazing. I went from knowing nothing about integrals to having a very solid foundation on integration. Thank you!

[@dexio85]

Have to catch a plane in 7 hours in the morning. KZbin recommends 6 hours of integrals. Guess I'll sleep on the plane :-P

[@hydropage2855]

Thank you very much for this video, professor Chow. For the last few days I've been sitting down and figuring out every integral by myself before you did it, and I've finished. This was a very good experience for me. Keep doing what you do, you're invaluable

[@blackpenredpen]

I am happy to hear this. Thank you and best wishes to you 😃

[@arturokullmer8634]

You are the only person that makes me study and learn in holidays and free time. You are a very good teacher, and i think you motivate a lot of people to study math. Thanks you. :)

[@Rocky.7707]

This is what KZbin is for. Thank you so much brother for helping all of us. God bless you.

[@mingbeckwith2232]

I'm taking Calc 2 right now and I went from absolutely hating it to feeling like "I got this" because of your videos. The way you explain things makes it SO much more approachable and I ❤️ your sense of humor. Thank you thank you thank you 🙏

[@blackpenredpen]

You got this!

[@mingbeckwith2232]

@@blackpenredpen I'm watching this video rn studying for my midterm. Thank you for the encouragement you're so nice 😭

[@Baerchenization]

Just remember this: next time you go to McDonald's know that you are the only guy in the room without a Phd... :)

[@blackpenredpen]

: )))

[@totallynotwei6830]

I didn't know my dog had a phd.

[@anonymaus8191]

Funny.

[@chronicsnail6675]

Nerds think that all there is to a man is the letters behind there name...

[@a_spire]

Pizza Hut delivery?

[@kushagragupta8068]

The amount of blessings and wellwishes you must have garnered from students across the world must be phenomenal. You're doing God's work, sir. Thank you very much.

[@erforscher]

I will watch this in place of movies. Thankyou very much Sir. From an Indian student.

[@blackpenredpen]

Thank you!!!!

[@ncognito12]

Thanks for this video bro, without your help I don't think I would have understood integrals at all. 😊

[@blackpenredpen]

You’re welcome. Thank you for your super thanks, too.

[@waainima]

I’m on Diff Eq right now, but you save my ass back then. Thank u, much love from the Philippines!

[@Vandarte_translator]

good job

[@GreAse0MonKey27]

Best Moment 1:33:10 (edit and 3:06:17) That's Dedication. Teaching whole integration to a camera in empty room on a white board while holding a mic for 6 hours straight. Thank You a lot for this video. More edit More Funny Moments --> 3:39:10 "Welcome to the theta world" 3:39:42 "Tangent square theta" 3:40:00 "how to make 1 to 9" "Don't say plus eight" :D

[@MadaraAzukar]

Tan² theta is best

[@paulhaso]

Ok I'm not learning this yet but you must be a hero for SO many people right now.

[@blackpenredpen]

: ))

[@SupremeDragonEmperor]

I'm studying for my calc final with this, and it's been *SO FREAKING USEFUL* thank you blackpenredpen!!

[@lightningmcqueengaming2038]

Same here brother. Good luck!

[@grahamwilson8843]

I have slowly but surely been falling behing in my calc 2 class, and this is giving me some confidence that I'll be able to get back ahead of it before too long. Thank you so so much!!!

[@mariialeonn]

im using this video to study for my calc 2 midterm, which covers all the topics in this video! you make integration so much fun!! thank you!!

[@wrighty1729]

I can see passion in your eyes. Respect.

[@blackpenredpen]

Thanks!

[@haroldvillacorte2092]

21:30 I like the way you did U-substitution, solving for dx then substituting it back in. Up until now I've been reconciling it my head through some type of strange osmosis.

[@DerLeeker]

great for a sleepless night, more interesting than reading Kant for 1 minute

[@DerLeeker]

this is a bit ambigious ... i meant by that that i could probably hold my concentration for 6 hours straight when doing math (the best drug there is!), but would loose it after 1 minute of Kant

[@CleverMonkeyor]

More interesting than reading Thomas Aquinas for 10 seconds.

[@strafe4922]

This gentleman really knows how to teach math!!! I wish he was my Calc II prof right now he's made this much more clear in 5 hours than 25 hours worth of lectures with my prof!!!!

[@marty5585]

Thank you so much for this video! Before watching this I had a bad understanding of the integration techniques and I wasn't sure how to do most of these. After finishing this video I feel a lot more confident about my upcoming exam thanks to you. I appreciate you spending the extra time explaining the steps.

[@keronasr6547]

This man either started this video right after his work shift or was ready to start his shift right after recording. Huge amount of respect for your work ethic and dedication. You've helped me throughout many exams !

[@botondzalavari4209]

I'm going to 11th grade in Hungary, and I'm probably not going to use this until university but I like math so I study it in my free time as well. Thank you BPRP for making amazing content!

[@KoopstaKlicca]

I've been out of school for a few years and I was put immediately put into a Calc II class, so I basically have a 5 year gap between Calc I and Calc II class. I've been having to spend like 4-5 hours per night after work to just catch up and relearn a lot. These videos aren't just educational but motivational to see someone spending as much time as I do in a session. Thank you so much

[@johanvanwyk9017]

writing Engineering maths in this week, I watched the whole video and did each example with you. It's clear that I had to learn my derivatives before trying to do the integrals myself. thank you for saving my module!

[@craigdean5013]

Liked your comment thanking your prior instructors who mentored you. So often most of us thrived because of a similar mentor earlier in our educational journey. Your working knowledge of mathematics is electrum. I am watching your podcasts daily to engage my mind more as I age. I am seventy five and using your workshops to minimize dementia. I am a physician but studied engineering over fifty years ago. It is fun to try to rediscover the integral calculus of my earlier life. Nice job.

[@beterax]

I love this video so much, I have a big test tomorrow and this is like the ULTIMATE review of every method. Thank you so much!

[@elquesohombre9931]

Actually surprised myself seeing as how I am watching the entire thing wthout skipping through sections or anything. It's genuinely great review and for those who have never learnt much about integrals, this is a great way to get into them. 4 years and still incredible.

[@danielrinconcaicedo9272]

You just give me the courage that i lacked to really study integrals, THANK YOU.

[@blackpenredpen]

Glad to hear!! Thanks

[@cuberdaboi2397]

Thank you for making this, I’m in 8th grade and study math on my free time. Your videos are very helpful and you are a great person for doing this 😊

[@rektspresso7288]

If you're in 8th grade and already studying math at this level, that is extremely impressive and just really cool. Keep it up!

[@blackpenredpen]

Derivative starter next?

[@abhishektyagi4428]

SIR the resources and links to learn mathematics that you said in you video with fematika are still NOT UPLOADED in the DESCRIPTION of the video , Sir please do upload those links

[@justabunga1]

Yep, let's do that.

[@Nickelicious7]

Could you do a Series starter at some point?

[@blackpenredpen]

@@Nickelicious7 ok!

[@abhishektyagi4428]

@@twopie6911 nothing just wanted to know the resources could you help me Don't say Google

[@zacbenson1743]

Just to express my gratitude, I ordered your integral t-shirt! Thanks so much for the help and dedication!

[@blackpenredpen]

Thank you for supporting! Best wishes to you!

[@me.5622]

This is one of the best math videos ever made.

[@sreevinayaks8027]

The amount of joy I got from the video realising I know 90 percent of the stuff is immaculate. Thanks for the video Math God.

[@zacbenson1743]

You and the organic chemistry tutor are single handedly backpacking me through calc2-3… is there any chance you can start a tensor calculus series?

[@alizachazin1063]

You retaught me integrals two days before my final. I cannot thank you enough

[@monica-ww8gi]

honestly integration is a lot more fun then people make it out to be

[@Nain115]

You do know. :3

[@vianneyedie4066]

Lebesgue integration? Fun? Lmao

[@doddlebutt5352]

@@vianneyedie4066 99% of the people are referring to Riemannian integration... but you're right, Lebesgue integration is the worst.

[@Azrail127]

Indeed

[@LORD-kg4sl]

@@doddlebutt5352 both are fun when you know to do it i dont know how to do it though lmao

[@bernard00g]

Excellent!! I used this video to study with my daughter and was all we needed. Thank you!!

[@TheSpokenBanana]

Thank you so much for doing this. I'm taking Calc II right now and it's been going pretty rough. I never got a real good understanding of integrals in Calc I. But this video has been super helpful to me. Thank you again! Hopefully I'll pass the class!

[@kristianhans7588]

Just finished the whole video and I have my final exam tomorrow morning. Only 6 hours to go till my final calc 2 exam. I have faith that I'll pass. Thank you so much.

[@kristianhans7588]

@@shrewd69 it was okay...I'm not gonna lie that I knew everything but I was able to do several integrations. The thing is calc 2 exam is not only integrations but also word problems and applications...so for the integration parts, they went well 🙂...hats off to blackpenredpen🫡🤝

[@shayenkerai8853]

I'm still studying A-Level Maths so this should be too advanced for me but I watch so many math videos in my spare time that I managed to understand every problem. Thanks for explaining everything so simply when there are complex ideas at play.

[@muzamilyasin8360]

We do this in a level maths though.

[@kaamilmirza6963]

This is such a life savior. I pause video on questions, try it myself and if I am stuck or don`t get them right, you are there to explain. Thank you.

[@georgepeterson2655]

Just watch this impressive Math channel kzbin.info/door/ZDkxpcvd-T1uR65Feuj5Yg

[@kaamilmirza6963]

@@georgepeterson2655 thanks man sure

[@shreyaskelkar2057]

Big fan of yours..... I'm also a student from India appearing for IIT-JEE exam this year..you have helped me a lot by providing quick tips for calculus Thank you sir 😊✌️❤️

[@amoghgirgaonkar9512]

Haha gotcha

[@shreyaskelkar2057]

@@amoghgirgaonkar9512 o..m..g!!!!!!!

[@cm221b]

Kaha pe help kiya re

[@roejogan1759]

@@cm221b arey wah

[@kezmichničano]

I am an incoming engineering student and I am very thankful for your videos. I hope all educators were as dedicated as you in teaching!!! (hoping to pass calculus 2)!!

[@Neo-Anderson7]

Thank you so much. From what I have seen on KZbin, you are the best math teacher. There is never a dull moment in your videos. Your work is greatly appreciated. 🔥🫡

[@kamarieirina]

This is my 1st video i watched on this channel , 1h in and im in love , the structure , the explanation-amazing work and effort , im gonna say , if teachers in schools / uni would put at least half the effort you did , we would most likely be all mathematicians 🙏👏👏 thank you kindly from all of us , for putting everything on shelves in our minds 😬 i will watch this one and more of your videos and im sure ill do better on my exam 💪Thank you again🙏🙏

[@blackpenredpen]

Glad to hear! Thank you for your kind words!

[@mubianalisulo2200]

Simple by, looking. When someone is explaining for him to be able to elaborate well like this he has past through thick and thin .you are doing a very good job sir keep it up .

[@manav8094]

I am preparing for jee mains , which is a testing exam for students aspiring to do engineering . And yes i could do all these questions you did ... but the way you did was awesome . Without any break . Wow !

[@blackpenredpen]

Thanks!!

[@naloliwaerasmus9929]

This maybe 4 years old but it really is helping me, thanks alot I have been looking for this for long

[@maulanayafiedanendra431]

I really enjoyed this. Brings me back the memories in high school back then. Your explanation is clear and fun at the same time. Good job!

[@row3186]

You make things so much easier to understand. Very straightforward no nonsense style of teaching.

[@whoami8434]

I literally hated all three semesters of calculus but somehow you make it sound fun. I hope you have videos for differential equations.

[@blackpenredpen]

Who Am I? I have them already. Please check my playlists

[@whoami8434]

blackpenredpen dawg, you the best. Much love from Ohio.

[@michalisl.3798]

I'm studying electrical and computer engineering at the university of Patras, Greece. Thank you for this. You're an amazing teacher.

[@talramot2631]

It's a really helpful video, i couldn't stop watching it so I watched the whole video!

[@blackpenredpen]

Excellent to you as well!

[@Ahmad_Fawaz]

Well done man We appreciate that 🙏🏻❤❤

[@cameronfraser709]

1:42:23 Happens every other week.

[@blackpenredpen]

lol!!

[@pokerboy72]

Hahaha

[@okworld.____2187]

You're the version 2.0 of the mathematics legend Mohit Tyagi sir from India. appreciate your job of amalgamating all those integration techniques together, much needed lec. thank you!

[@NikitaNair]

I can literally watch your videos for hours.... You are so knowledgeable and you really grab your viewers' attention ❤️ Sending you so much love for teaching Mathematics in a fun and interesting way!!!

[@loubeauchesne8771]

was procrastinating studying for my calc 2 final and this came up in my KZbin feed, a very good refresher and clarification on a lot of material, thanks :)

[@panavsingh4846]

Best maths teacher I have seen

[@AndyGoth111]

1:41:55 I'm pretty sure you don't strictly need complex numbers to factor x²+4 = (x+2√x+2)(x-2√x+2). Just do this: x²+4 (x)²+2(x)(2)+(2)²-4x (x²+4x+4)-4x (x+2)²-4x (x+2)²-(2√x)² ((x+2)+2√x)((x+2)-2√x) (x+2√x+2)(x-2√x+2)

[@mathemagicianpi7526]

Andy Goth I believe factoring means to write a polynomial as a product of two other polynomials. sqrt(x) does not count as a polynomial.

[@AndyGoth111]

Fair! Thanks for the clarification

[@sarojpandeya9762]

I am recommending my students to learn from your fabulous work. Greetings!!!

[@blackpenredpen]

Thank you!! I really appreciate it!

[@Arutiwari_18329]

Just watch for enjoyment..really good❤

[@josephymenchu]

You are such a great inspiration. I aspire to be a teacher like you.

[@loopilicous]

I cannot believe the time and effort you put in to this video (and others). It is so extremely helpful. I am in engineering and did not have a good Calculus foundation which has bit me in the butt. Thanks to you, I now can master it so integration does not worry me as much! Thank you for your passionate, clear, fun, and helpful teaching!

[@danielclarke8053]

You, sir, are the best teacher I've seen in a while!

[@pseudolullus]

It's been 15 years since I passed Calc 2 and this video has been excellent reviewing material!

[@yuganshujain691]

Integral of √sin(x-a)/√sin(x+a ) ..a good example of substitution..

[@rjfontenotiii]

I need to keep practicing. I am starting Calculus 2 soon, and it took me about 15 hours just to get through the 36 integration by parts problems at the end of the chapter in my textbook. I am getting better, and I'll keep drilling those integrals until I get it down. You make this look too easy. Thanks for all of your videos!

[@NileshRamphal]

I always watch your videos when I need inspiration to learn maths and calculus.

[@annexsoftworks]

Liked, subscribed, rang the bell, downloaded the PDF and commented lol it's the very LEAST I can do for all your hard work!

[@blackpenredpen]

Thank you!

[@annexsoftworks]

@@blackpenredpen Mid terms tomorrow! Currently sitting at a C+/B- so just need to get through tomorrow! Post-bacc. student so it's been about 8 years since Pre-Calc. currently at UNO for CS.

[@annexsoftworks]

About an hour and a half to go :D Watched many of your vids in Calc. I, and pulled through with a B but that class was a lot harder!

[@rurafs7934]

Thanks you teacher ! I was able to understand calculus on 10th grade, usually differentiation and integration is be taught on 12th grade, but because of your channel, I was able to understand it within 1 year! And one more time, Thanks you teacher

[@RyanForx]

Ay we almost have the same time for understanding calc, except im in 9th grade and started learning some of calc 1 at the end of 8th grade, good luck

[@neilgerace355]

4:13:10 From 1982-85 I went to Swanbourne Senior High School, where I studied some calculus, I think it was about your Calc 1 level. We had to learn all the trig identities under the sun in Year 8 or 9. The hardest one I still remember is: 2 cos A cos B = cos(A+B) + cos (A-B) which we learned as 2 cos cos = cos sum + cos diff. However, watching your videos reminds me of so much I'd forgotten! I enjoy learning it all again from you. Cheers from Perth, Western Australia!

[@blackpenredpen]

Neil Gerace ah nice! I am happy to hear. Thank you for sharing, Neil!

[@ZephyrEclipse]

Fantastic job! This really helped, thanks for putting in so much effort!

[@blackpenredpen]

Benjamin Sadai my pleasure to help!

[@EthanSimmons06]

1:36:00 Difficult: Something that is labor intensive and has a high chance of failure Easy: Something that requires little labor and has a low chance of failure

[@diegosc985]

Thank you professor, you helped me quite a lot. I think I love calculus thanks to you. I'm at George mason university in Virginia.

[@blackpenredpen]

Glad to hear! Best wishes to you 😃

[@d-rex7043]

Thanks again BPRP - Been watching the Penn marathon and doing each of these sections at the relevant place, as a refresher for the next semester. Starting to feel match fit!...

[@nadiapaoladejesusramirez5688]

Wuo you're incredible! I admire you !

[@blackpenredpen]

Thank you!!!

[@nadiapaoladejesusramirez5688]

I love your videos, you never stop doing it, please