Integration Using The Substitution Rule

With the basics of integration down, it's now time to learn about more complicated integration techniques! We need special techniques because integration is not as straightforward an algorithm as differentiation. The first technique we will add to our bag of tricks is the substitution rule. Check it out!
Watch the whole Calculus playlist: bit.ly/ProfDave...
Watch the whole Mathematics playlist: bit.ly/ProfDave...
Classical Physics Tutorials: bit.ly/ProfDave...
Modern Physics Tutorials: bit.ly/ProfDave...
General Chemistry Tutorials: bit.ly/ProfDave...
Organic Chemistry Tutorials: bit.ly/ProfDave...
Biochemistry Tutorials: bit.ly/ProfDave...
Biology Tutorials: bit.ly/ProfDaveBio
EMAIL► [e-mail]
PATREON► / professordaveexplains
Check out "Is This Wi-Fi Organic?", my book on disarming pseudoscience!
Amazon: amzn.to/2HtNpVH
Bookshop: bit.ly/39cKADM
Barnes and Noble: bit.ly/3pUjmrn
Book Depository: bit.ly/3aOVDlT

[@williamgutierrez9227]

Professor Dave is the teacher you always wanted but never had. Thank you, Professor Dave. You have saved me multiple times.

[@abdallababikir4473]

I have a great teacher soooo...

[@omarel-nemr6506]

+1

[@spike35540]

You just saved me another panic headache. You explain things so well, even my engineer of a father recommended you over his own explanations. Thanks a lot!

[@Syntaxxed]

my new study method is watching 10min youtube videos before reading my text book. Idk how they do it, but each and every teacher/text book manages to make me super confused and make everything harder than it actually is. Thank you for your service prof Dave!

[@kavyahegade6477]

There is a simple way to understand science.. Just take your phone/pc --> youtube --> Professor dave explains.. that's all

[@hafsarazzaq929]

He explains so well in so little time. I love the excercises at the end because it helps me review what you taught and makes me more confident in what I learnt.

[@samberg7400]

Dude my AP calc exam is tomorrow and no one could explain substitution to me until you, I’m in debt and at your service!

[@is6815]

I'm a 12 year old tryna learn calculus and also I'm from the philippines and glad to say,this helped me a lot!
+1 sub for you!

[@38Fanda]

You can learn all high school, college math within a year by giving yourself a reason to do it for, since the brain is evolved to do certain tasks, for survival and reproduction. Save yourself a lot of time and money by doing AP exams way earlier, screw the norm brother.

[@bologna3464]

Damn, respect to you. I'd never be able to understand this at 12.

[@BossKing1243]

@@bologna3464 facts lol

[@add852]

saan ka nagaaral?

[@wizard.01]

You're wasting your childhood

[@joshuayapasi4335]

Man you're genius. Every problem just make sense.

[@Dius21]

Thanks a lot!!! I usually have no problem in maths but for some reason I simply wasn't able to wrap my head around integration by substitution. This really cleared things for me and lifted a huge burden!!

[@ameliab7753]

I stan this man and will forever thank him for existing i can already feel my calculus 2 grade rising

[@richardbroder4784]

Dave - all I have to say is THANK YOU SO MUCH. This video was extremely helpful in developing my understanding of the substitution rule

[@samcoopermusic]

Dear Dr. Dave, I would hereby like to submit a humble request for your consideration of the production of an additional calculus video, U-Substitution of Definite Integrals. There's some extra thinking involved, and I would love to see your 5-10 minute take on it. Thanks for all the amazingly helpful content!

[@daaaaniiiiii]

gotta love professor Dave, finally understood the concept after a WEEK of working on this!! gotta love him :D

[@Jupaaa]

Jesus you look like dave

[@BoothPlayer]

You're an amazing teacher! Pls keep up with the great work!

[@ivoryas1696]

I'm sorry, I know this is said often but... holy cow. This man is up here explaining things that, a year ago, I would have understood about as well as Einstein's Field equations and yet I'm understanding it. The crazy thing is that even though it still sounds nerdy af (which is something I always used to appreciate, but wasn't really able to properly replicate due to a lack of understanding), it's not just going in one ear and out the other.
Now I'm not gonna say that it's solely Dave or even just solely CZcams, but between last year and this year I went from failing at properly comprehending Pre-calc class to sucking in information on Calculus and planning on taking an exam to skip taking the first class all together, and I feel like it wouldn't be possible at *_nearly_* the speed it happened without CZcams and Dave more than many. I could even say even Khan Academy hasn't helped as much.
Thank you, Professor Dave. I first came here for anti-conspiracy content... but I subbed for this.

[@TheBrammiej]

This actually helped me a lot! Thanks!

[@entertainmentts4090]

czcams.com/video/SbP3_d15yFc/video.html

[@CloroxBleach-rr8gr]

This guy is genuinely better than organic chemistry tutor. Things are always so clear and straight to the point.

[@bonelessbooks9263]

I won’t stand for this Julio slander

[@laxmishachandrashekar7187]

truee both are good@@bonelessbooks9263

[@user-uh1gz9by7u]

You are very wicked, organic chemistry tries his best,but you can't appreciate him

[@jrsun]

Giga god, i thought d/dx was just the notation as well. Thanks for not skipping over that explanation like others.

[@_56trapped32_]

Sir you don't even know how much of a headache and heartache you saved me from ❤️ 😭

[@umamarajput456]

Your way of explanation made things pretty much easier

[@shreeshkulkarni6953]

Professor Dave is my gratest teacher ever I have seen.I love your teaching methods

[@hamzaalaudi1184]

God bless you professor Dave, you are saving lives out here, emotional lives that is

[@firewarrior9999]

I needed someone to explain this to me like I was a child.

[@kw_2707]

I FINALLY GET IT!!!! I have a test tomorrow morning and substitution was the only thing I couldn't grasp. That passing grade's in the pocket for sure

[@kw_2707]

Thank you so much Dave!!! I got a 100%!!

[@luyandamiddleton2069]

thanks Dave. I am preparing for my final exam for one of the hardest math courses!!! tnx

[@faithsmith158]

Do you have any advice for me learning calculus like in the video? I’m in high school and not doing too well :( . It’s hard for me as I have to juggle many subjects.

[@ProfessorDaveExplains]

Just start at the beginning of the calculus playlist and work your way through?

[@entertainmentts4090]

czcams.com/video/SbP3_d15yFc/video.html

[@SamArunothayan]

Thank you professor

[@agepallysaiteja]

You saved my life.Thanks that you are born.

[@erikaespina]

THANK YOU THANK YOU THANK YOU‼️‼️‼️‼️ i cant believe that all of my questions abt the substitution rule have been answered with a short, 10 minute video!!! THANK YOU‼️‼️‼️

[@Dcoooooooookkkkooo1]

Calculus is life

[@adityaram6845]

The best teacher on CZcams.

[@k.g.vishal6690]

Love From INDIA❤🇮🇳‼️

[@thebeast1383]

math with this guy is on another level

[@kamy724]

I needed this like 3 years ago

[@another7920]

It's not too late to learn now

[@mountainking1166]

Thanks! I came for help with organic chemistry and stayed for the calculus help.

[@nikedanmechatronicsengnr845]

Another best teacher across the world is our respected sir dave. Sir i am from india and still my all my country mates watch ur chnnl to ace science.

[@ave1687]

I finally understand my lesson in calculus. Thank you!

[@aarya_dils]

Thankyou❤️

[@punditgi]

What is the video in this series that is where Dave justifies the splitting of dx from d/dx to be able to "multiply" the integrand by dx to get 2x dx = du?

[@malasinha6144]

Thanks prof. Dave

[@drvir]

Thanks a lot!

[@ajlpz]

Thank you Prof Dave!

[@kate5406]

finally a video that explains this well.

[@EugenMarianPopescu]

What have I been doing in high school? I didn't learn anything back then. Now, I understand everything. Where were you Professor Dave when I needed you?

[@vinzusdergraf8581]

Professor Dave, please can you start a series on DIFFERENTIAL EQUATIONS?? Pleeeeaasseee??

[@ronanshiroshi6920]

Very amazing technique that can be shared to my classmates 🤗

[@ALixKilleR0]

how do you spend your day ?

[@ronanshiroshi6920]

I'm spending my days this summer self studying in advance for the nxt school year

[@aselim20.]

I watched third time and I took notes.

[@daralla3272]

simple and easy steps to understand thanks Man

[@user-lm4ly8cu9f]

professor dave explains is so cool

[@horizonbrave1533]

Hey so is it possible if you have an extremely long polynomial, that you can use the Subsitution method in two places in the polynomial at once? So if you had like two sets of parenthesis in a long poloynomial, could you substitute one for U and another for Y (random variable), and then do this method for these two different substitutions at the same time?

[@ave1687]

I always go back to this video whenever I can't remember how to do u substitution. It's been almost a year already

[@shonaru2223]

Thanks very much Prof. Dave!
For helping me to understand it more.

[@painnagato1825]

Thanks a lot sir, after long time of learning I've understood by yuor explanation

[@choukriazzouzi5796]

U should say that x ≠π/2
Bcz if x=π/2
Than cos π/2=0
ln0= unlimited form

[@yungmaz13]

Is the point in the substitution rule to simplify the equation? Cause if so i dont really see it it just makes everything more complicated than it already was. Either way, great video 👍👍

[@entertainmentts4090]

czcams.com/video/SbP3_d15yFc/video.html

[@yeehaw5351]

I'm totally using that smiley face variable on my next exam

[@Asifkhan-zx7wi]

this is the best way to learn mats

[@simar4723]

For the second example that was shown, after applying u-substitution for it to then become du/3=x^2dx; what would happen to the x^2?

[@9308323]

It (and dx) will be represented by 1/3 du. It's already part of it.

[@user-gb2xp3ip1b]

Thanks a lot professor ❤❤ love from Bangladesh

[@ayushnirankari7304]

He teachers so cool man thanks!!

[@moniqueabigailcarballo3518]

Hello, professor. I am just confused about the true answer in the integral of tan(x), if it must be ln |sec x|+c or -ln |cos x|+c? Thank u.

[@aceganot6598]

right?

[@danielrosen4496]

Thank U, Prof. Dave.

[@DrakeRing]

Thanks a lot

[@wilfu2050]

Unajua mwanangu // awesome 👌

[@sahilraj9554]

thank you sir

[@jericholegend447]

Thanks sir!

[@silentintegrals9104]

Great Video! Solving integrals is allways fun!

[@user-hu4jd7hr4f]

9:43 The answers are in order: ln|x^2+3x| + C and -cos(ln x).

[@ak_oneoneone]

why is there a minus beside the cos?

[@user-hu4jd7hr4f]

⁠@@ak_oneoneoneBecause the derivative of cos is negative sin.

[@ak_oneoneone]

@@user-hu4jd7hr4f ohh yah that makessenss

[@taimoormanzoor7386]

He is simply explaining the hard thing and God bless you.

[@priyank5161]

I watched a 30 min long video on yt... Not gonna take his name... And I was completely lost
I couldn't understand a shit
And then i stumbled upon ur video... Seriously everything is easy now... A bad teacher can make u feel dumb.. really
Thanks dude for telling me the correct path when I got lost with that random youtuber

[@srividyaraja1454]

It helped me

[@razasyed575]

an amazing explanation brother

[@Felix-uw8nv]

Thank you!

[@nwaosman6785]

Thank you best professor 🥰

[@seanrhettzamaylanatividad6213]

shoutout sa mga pinoy dito HAHAAHA thank you sir dave!

[@AlejandroDelaTorre-hd2lq]

Thank you Math Jesus

[@priyalsampath7192]

thank you

[@johnchang1226]

Thanks very much.

[@addisufiapostolicgeneratio2093]

Thanks

[@ak_oneoneone]

second number in evaluate, why did the f turn to F(x) and in turn getting a negative sign beside the final answer? appreciate the help

[@RishiKtr_IITMandi]

You are doing a great job...keep it up

[@starstruck7086]

Professor dave, I have a question, is the U substitution the same as algebraic substituion?

[@carultch]

It's called "U-substitution", because it is common to use U as the temporary variable for it. I prefer to use W instead of U, when both integration by parts and U-substitution are part of the same problem. It is not the same thing as algebraic substitution, but the term substitution means the same thing in both of these terms.
Ultimately what it means, is if you can recognize an integrand that is likely the result of a differentiation through the chain rule, then you can switch the variable of integration, to a variable that represents the inside of a function. Suppose your integrand is a product of g'(x), and the composition of functions f(g(x)). The g'(x) term might not strictly be the derivative of g(x), but rather you might have to multiply by a constant to get g'(x). In that case, you multiply by 1 in a fancy way, as a constant divided by itself. Part of that ratio becomes the constant you need in order to complete g'(x) to be, and the other part of that ratio, becomes a leading constant of the entire integral, that you pull out in front.

[@ian.ambrose]

@@carultch Didn't even give you a thank you. What an asshole they are.

[@nicholasdingle6805]

I love you professor Dave

[@mehrabanbradosti1888]

you are intelligent thank youuuuu

[@mutaharamasiatmohona2946]

Thank you so much 🥰 that was very helpful 😊

[@aelafanteneh2767]

Thanks a lot brother

[@The_Bad_Ant]

Thanks a bunch 💐🙏

[@esthertesfai9559]

thank you so much!

[@ankitaaarya]

2:15 where did you justified ehy its possible??????????????????????

[@ehatipo4598]

Thank you sir! :)

[@cristinageorgescu8137]

Thank you😊👍

[@joelvarela7785]

Thanks professor, now calculus seems reasonable, greetings from Mexico

[@cloud_222]

In which video of the series was it explained why we can manipulate differentials here? Thanks

[@alkasantosh9012]

Thanks sir
It helped a lot

[@silentintegrals9104]

Great!

[@tGoldenPhoenix]

Done.

[@tGoldenPhoenix]

Comprehension done.

[@punditgi]

In which video does Dave tackle u substitution of definite integrals and how to change the limits of integration?

[@carultch]

If it's my choice, I leave the limits of integration alone when doing u-substitution, and translate it back to the original variable of integration, before applying limits. Some integrals are not possible to do in closed-form as indefinite integrals, so that is the rare times when I opt to translate the limits to the u-world. For instance, the Gaussian integral.

[@brysonmutinda5280]

Professor Dave makes math look simple 😄😄

[@user-hv6ef9ie1g]

8:27 Shouldn't the minus sign become plus for the left part of equation, if we add it to both sides?

[@mamalgolabi7666]

You are good