What Integration Technique Should I Use? (trig sub, u sub, DI method, partial fractions) calculus 2

[@blackpenredpen]

If you guys like this video, give it a like and share it with others! Let me know if I should do a part 2 for this? Here's the file: docs.wixstatic.com/ugd/287ba5_f1afe86da0454ef08c92eae9b9e9c18c.pdf Here's the playlist to the solutions to everybody kzbin.info/aero/PLj7p5OoL6vGwS5ilte_7fjRB2rQ_yE_Hj integral of ln(x)/x^3, 0:45 integral of sec^4(x), 2:29 integral of (2x+3)/(x^2-5x+4), 3:50 integral of x^2*tan(x^3), 5:50 integral of 1/(1+x^2)^(5/2), 7:17 integral of e^sqrt(x), 8:50 integral of sin^2(x), 10:37 integral of 1/(sqrt(x+1)-sqrt(x)), 11:20 integral of e^x/sec(x), 13:15 integral of 1/(1+cos(x)), 15:09 integral of (x-4)/(x^4-1), 17:44 integral of x^2/sqrt(1-x^2), 19:24

[@iluxan5648]

omg, thank you very much for this video, I just had this kind of lesson today at my math class, I can learn better now. Also I like everything you do so keep it up ^^

[@blackpenredpen]

iluxan 9_6 Thank you very much for your comment. Don’t worry, your English is good!

[@alcmaeonofcroton5060]

Please do another video, maybe going over the methods and their basics, my calculus teacher is really bad at explaining them and your videos help a lot!

[@rithikkapoor6465]

@@blackpenredpen try integrating x^(x/ln(x)) in your next video. You will be surprised by the answer.

[@redforddonguya8690]

PLEASSSSSSEEEEEE DO!

[@ABruckner8]

This is probably the most important Calc II video ever produced. It focuses exclusively on strategy, which is the most important element in all of calculus: Learning strategies (creativity), and recognizing when multiple strategies are valid, and why some might be better. Bravo, Steve! It's similar to your D/I "When to stop" video. Ever since I've stumbled on your channel, I've been re-learning calculus just for fun!

[@blackpenredpen]

Bob Higgins Wow, what a comment!! I am very happy to hear this, thank you Bob!

[@bartoszbiaas3197]

What video are you referring to? I cannot find it

[@sambawithswag]

He meant How to Do parts Through D/I Approach .. Which was really Great!

[@obo190]

factsssss

[@suissetobacco]

Wholeheartedly agree. After developing a proper understanding of the theory behind integral calculus, students should focus on consolidating their integration techniques. As repetitive as these drills may seem to be, they can save a lot of time in the long run.

[@neeraj8278]

What integration technique should I use? Me: Wolfram Alpha

[@blackpenredpen]

abvsefhidj s best

[@hungryfareasternslav1823]

Wolfram Alpha is an idiot because it cannot calculate tetration...

[@johannesh7610]

Also, it cannot even understand the differential equation thatbI came up with: y'(x) = y(y(x)). It's quite cool though

[@mihaip1179]

@@hungryfareasternslav1823 how about super logs?

[@kaddanki096]

or you can try symbolab lol. I always randomly type in differential equations in there. Mathway is pretty good too.

[@MisterTutor2010]

Integral Calculus can be summed as "What Kind of Magic Spell to Use?" :)

[@manojkumarsahu7929]

kzbin.info/www/bejne/oHipkGmZeL6Jras Tough integration

[@masterjif9506]

Must be nice having graduated Uni. Now it’s our turn to use the magic spells

[@CliffSedge-nu5fv]

Calculus does feel like magic; that's why I love it.

[@spelunkerd]

This has to be one of your best videos, it really made me stop and think. The ability to stop the video, think the problem through, then resume is far better than classroom teaching of the past.

[@blackpenredpen]

spelunkerd thank you!!! I am very glad to hear it!!

[@0100hem]

true. this is one of the best advantages over ordinary classroom.

[@manojkumarsahu7929]

kzbin.info/www/bejne/oHipkGmZeL6Jras Tough integration

[@Hester_prynne]

not really. i have questions, who can answer them for me? ofc i would have to wait for responses. then if i dont understand the answer, im still waiting on a response. a teacher could explain instantly

[@chandranichaki9580]

kzbin.info/www/bejne/kGeaq35qd5uWnLc

[@josammarenye4021]

This guy is hilarious. He makes calculus look like primary mathematics. Can't thank him enough 👏👏

[@holyshit922]

For the fifth integral there are two u subs which work nicely 1. u=x+sqrt(1+x^2) 2. u=sqrt(1+x^2)/x Integration by parts also works for this integral

[@MatthewChenIsAwesome]

"Take it to the u world" is now my favorite math phrase

[@gosth81]

They should make a t-shirt with that quote lol

[@siddharthagupta8528]

Underrated comment

[@Sajee_Venus_Aadheev.T]

Please tell the time when he say

[@Faith-zp5kf]

@@Sajee_Venus_Aadheev.T I think the person who made this comment thought of the joke = ).

[@ginnis_14]

@@Sajee_Venus_Aadheev.T 9:24

[@shhh9162]

his energy is so pure 🥺

[@nlight9214]

bro just saved me from calculus. I've always wanted to create a youtube channel teaching cool stuffs instead of hiding them. You are my inspiration

[@marrenroy4896]

This is great, my biggest struggle in calculus 2 has been that intuition you're supposed to have about what integration method to use. It does not come naturally to me, same reason the whole "guess and check" process of factoring quadratics was a nightmare way back in elementary school

[@sergey1519]

If you have a quadratic like x² - 7x + 10 = 0 we know that the two roots r,s satisfy r + s = 7 rs = 10 so what you can do is systematically consider all factors of 10: 1 x 10 = 10 but 1 + 10 ≠ 7 -1 x -10 = 10 but -1 + (-10) ≠ 7 2 x 5 = 10 AND 2 + 5 = 7 if 2, 5 didn't work you would then check all the other pairs of factors. Of course, you might notice that 1+10 = 11 and realize that you don't even need to consider (-1)+(-10), or realize that once you checked 2x5 that's the same as 5x2. In the same way with integration techniques you can systematicaly consider them one after another, just in your head trying to do the first steps, and once you find something that you seem to be able to make progress with, do that.

[@PenandPaperScience]

For those that want more practice, I have several videos where I solve complicated integrals step-by-step, using all the methods explained here. :) It might help some of you out there!

[@johnbiluke8406]

Nice!

[@PenandPaperScience]

@@johnbiluke8406 Thanks! :)

[@kendykeineele2843]

i am writting my exam this month ...its an engineering maths (city and guilds 401)'......can you help with the examples relevant to the syllabus

[@anjanasahu8416]

In 10th question 15:30 we can also use 1+cos2x= 2(cosx)^2 and then integrating (secx/2)^2

[@michaelsergi723]

Honestly, as fantastic as my TA is, you are amazing. Both of y'all have done an amazing job at explaining and writing out problematic scenarios that give us lower level students the biggest issues. However you present these scenarios in all of your videos so well. Thank you very much. (note: calc2 student)

[@kepler4192]

in 16:03, you can just use trig identities and turn "1+cosx" to "2cos^2(1/2x)", which you end up with 1/2sec^2(1/2x) and it's easy from there

[@nolentitus2120]

i throw the term lifesaver around a lot. buy me a morning coffee? lifesaver. drive me to campus when i'm out of fuel? lifesaver. but you my friend, you are the only one for whom the term lifesaver takes on its truest meaning

[@blackpenredpen]

I really appreciate your comment. I am glad that I was able to help.

[@abhishekpatil6071]

In the first question, we can rather make a u substitution Lnx= t and x = e^t 1÷x dx = dt So the integral becomes T÷e^2t dt Then we can use integration by parts.

[@blancaroca8786]

You used a t sub not u sub lol. And you end up doing an int by parts which is what the video does anyway. But I think your method is good! Thanks

[@FridgeGames101]

amazing video! really useful since I'm learning integrals right now

[@manojkumarsahu7929]

kzbin.info/www/bejne/oHipkGmZeL6Jras Tough integration

[@Shivam-wc3yx]

In the 10th question, you may write the denominator as 2cos^2(x/2) using the half angle identity and then write it as sec^2(x/2) and proceed from there.

[@letsseepaulallenscard.6604]

You are a machine, I’m very grateful for the daily uploads #YAY

[@manojkumarsahu7929]

kzbin.info/www/bejne/oHipkGmZeL6Jras Tough integration

[@owen7185]

You're one of the best KZbin has ever had

[@blackpenredpen]

Thanks

[@nikheetisnotcool2951]

This man is saving my Calc Grade.

[@teyhaleonard3982]

i have been looking for a video like this for ever ! I have not found one video breaking down the strategies you should use, which is the most important and confusing aspect of integration ! TYSM !!

[@roysupriyo10]

Actually in the first one, 'u' substitution is going to work. We substitute 1/x as u and we integrate to get (dx×-1/x²)=du. Then we take the natural log on both sides of the 1/x=u and get -ln(x)=ln(u). Then we can substitute all the values to get u×ln(u)du as the integrand. Cheers

[@shivanach45]

For the 6th question, Using u = e^sqrt(x) would give a faster answer. For the 10th one, using half angle formula and converting 1+cos(x) as 2cos^2(x/2) makes it very easy to get the solution.

[@King-sd5vg]

I’m so glad BPRP has such a big following, you always make me smile even when I see a very hard problem that I can’t fathom in your vids! I really want to become smart like this guy!

[@JesusMartinez-zu3xl]

you are literally like my cal 2 professor! and i love it! I watch these video to help in-force what i learned in class.

[@sollertia_]

In essence, try thinking about using trig sub: look out for single term square root denominators and 1+x^2 or 1-x^2 IBP: When you can easily differentiate one part and integrate one part PFD: When the denominator is a higher power polynomial or when the denominator can be factorized (pretty much the same thing, but not quite?) u sub: When the derivative of something can cancel out a troublesome part or when you're doing variations of a simple integration question but don't want to be careless (Feels like the most powerful and flexible tool in my elementary calculus toolbox, though, so probably has way more specific uses) I'm not in uni yet or anything so I'm not sure if my insufficient contact with calculus problems has led to some mistake of some sort in the summary above so please let me know if there are better ways or corrections that I would want to know about, thanks~

[@charleswoodard8478]

This channel has been a vital supplement to my studies!

[@blackpenredpen]

Glad to hear. Best of luck!

[@macscott956]

Wow. This video simplified so much for me. I wish I tuned into the channel ages ago!

[@blancaroca8786]

Nice revision! Ex.10 with 1/(1+cos x) use trig formula cos2A = 2(cosA)^2 -1 with x=2A to get rid of the 1 and leave just cos^2(x/2) on bottom.... basically now a sec^2 integral which is standard.

[@binodtharu8348]

Yeah i used the same

[@omermuhammet6526]

IN the beginning of the video he said that there will be some calculus 2 in this so I said I will concentrate on things I learn in calc 1 since I didn’t take calc 2. To my surprise I took every thing he explained in the calc 1 course in my uni. Now I really believe my uni is super super hard, I wonder what they’ll teach me in calc 2

[@ve4rexe]

for the first one you can just multiply and divide the whole thing by -2 x -2 and then shift one of the -2s into the power of x in lnx, after which u sub becomes very easy (u = 1/x^2, du = -2/x^3 dx) then youre left with 1/4 int(lnu du) which simplifies thereon

[@LauraUtter]

This video and the DI method video makes me feel like I’m learning something illegal! Haha I’m serious, you have no idea how much you’re helping me!

[@francis6888]

For #12 you could let u = arcsin(x) for the fun of it.

[@blancaroca8786]

That’s the same as sin u = x which is what he did except using a theta instead of a letter u

[@marchenwald4666]

I used another technique for the 10th First, observe that cos(x) can be written using Euler's formulas. Now you can use u substitution twice. If u = e^ix, you're left with the inverse of a polynomial, 1/(u²+2u+1). The result of this integration method is (1-e^ix)i/(1+e^ix), which's in fact equal to sin(x)/(1+cos(x)) (The last step needs some clean up using Euler's formulas and a readjustment of the integration constant to get a function that goes from R to R)

[@syedkhan8522]

You can also do it with the substitution t=tan(x/2) . Then cos(x) = (1-t^2)/(1+t^2) and dx = 2dt/1+t^2 . Everything cancels out and its much simpler.

[@marchenwald4666]

@@syedkhan8522 Looks pretty nice this way, indeed!

[@baharzubair]

shouldve had this on my recommendation two years ago. a life saving video

[@jeanjoelkapula5681]

I'm glad I've had a great calculus teacher . I can understand all this

[@alejandrotorres712]

21:15 "People don't like to put down a d-whatever. If you don't put down anything, that's like driving without your seatbelt on: very dangerous, so don't do that."

[@notizac_]

ive got a calc midterm in two hours, this has helped immensely, thank you so much

[@sultansuleyman6872]

for 10 task i have another way of solution, you divide cos x into cos^2(x/2) - sin^2(x/2), then 1 into cos^2(x/2) + sin^2(x/2). Then in the denominator will leave 2cos^2(x/2), plug in 2 in dx => d(x/2). Finally you get sec^2(x/2) which of the integral is ..... :)

[@humphibip]

My master, I don't know how I should express my feelings right now. Thank you very much for this video 🙏.

[@sassy-l8k]

his smile is way much appealing and makes session kinda more interactive :)

[@joschomo1010]

This was perfect. I have my midterm coming up and this was the perfect way to study as you give subtle hints without revealing the whole answer. Thankyou very much

[@akshaykishoredesai2017]

Ever since i found this channel, i started learning calculus and complex numbers with curiosity and for fun 😃

[@THE-LIGHT..]

This is a fantastic work,I'm pursuing integration and calculus 2 courses so actually your channel becomes my favourite. Wonderful job

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What you offer is a blessing for others

[@itsshahain5954]

Here's a very powerful method for integrating 1/1+Cosx . Simply use of t=tan(x/2) (btw if sin50x was present use half of the angle i.e t=tan(25x) ) and perform integration by substitution. Extremely powerful you can integrate 1/(sin99x + cos99x) , 1/1+cos100x you name it. Then sinx = 2t/1+t². And cosx=1-t²/1+t².

[@yernajdelavega]

I really learned a lot in your videos, especially the 100 integrals you solve. Aside from the fact that I like your speaking tone and explanation and lastly your eyes. So cute.

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I really appreciate and follow up ur tutor regularly. Wow i've got a lot of knowledge in easy way and surprised ur broaden mathematical wisdom. Mathematical difficulties of both derivatives & Integration become easy with ur lovely way of teaching methodology. Many thanks indeed. NA from Ethiopia

[@SirPlotsalot]

Well that's going to actually be some of the best revision I've got for my calculus exam in 4 hours :)

[@articcircleado]

Thank you for your kindness in making these videos. I love how happy you are and smiling, it helps me stay positive during math. Thank you!!

[@amberdeshbhratar9342]

In triangle ABC ,D is mid point of AB, E is a point on trisection of BC nearer to C . Given that angle ADC=BAE, find angle BAC.

[@Antonela.Stoica]

Happy Easter! Much health, happiness, peace of mind, peace. Thank you!

[@timothymagsino]

He’s literally so much better than any other teacher out there.

[@Zephyr-tg9hu]

This was a really cool video. I especially love how you integrated e^sqrt(x)!! Very clever!

[@travismora9046]

Great calc 2 video. Its a good refresher. Im going to write down these problems to have a set of problems to practice.

[@aleksandarmanojlovic4988]

for the 10. integral you could use substitution u=tan(x/2), it's a beautiful substitution that works every time in integrals where you have sin,cos,constants and their addition

[@abhirupbhattacharyya7491]

Best channel for learning mathematics..... U r the best😀😀😀😀

[@binodtharu8348]

Alternate solution to 10th q: use identity cosx=2cos^2(x/2)-1 and hence it would become super easy(1 step solution)

[@caiphusmogwaneng1466]

you really helped me a lot especially on that video with 100 integrals in 6 hours, may god bless you

[@blackpenredpen]

caiphus mogwaneng glad to hear!! Thank you!!

[@brickmotion6637]

For number 10 I used the power reduction formula in reverse: 1/(1+cosx) = 1/(2*½(1+cos(2*(x/2))) = 1/(2cos²(x/2)) Now set u = x/2, 2*du=dx, so you get the integral of (2*du)/(2*cos²(u)), equals the integral of du/(cos²(u)), which all in all is equal to tan(u)+c, plug in x/2 for u: tan(x/2)+c

[@sfundoy5dube59]

You're indeed Mr Calculus Finisher. You're the best Sir

[@MwalaBrooks]

My favorite guy who makes advanced calculus to look like primary school math 💫🔥🔥

[@nicholasesposito1204]

just took my calc 2 midterm and definitely did well thanks to your videos. Thanks!!!

[@sweartoyeezy3439]

problem #3 and trig sub is never talked about enough in calculus. Amazing video watched it twice!

[@mengnjoeusebia1335]

Please this video should be given an award as the best integration simplified video ever 👌👌

[@pinkninja2463]

for the third one you can just turn the numerator into the derivative of the denominator and split it. It's way more easier!

[@blancaroca8786]

Nice but We still need to integrate the split off bit which is basically 1 over the denominator. So we end using same method?

[@nico3727]

If I pass my Calc 2 class with a B its because of you! If i get my degree then you will get a donation for helping me out more than my own teachers.

[@krishgarg2806]

for 10th one, we can also use 1+cosx = 2cos^2(x/2) which will give sec^2(x/2) which can be easily integrated.

[@MorDikEye-n-Rigby]

On number 1, A u sub of ln(x) would mean x = e^u, and du = 1/x dx. that would leave you with u*e^(-2u). Then integrate by parts!

[@vladislavziyangulov7490]

For the 10th one: you could also say 1+cos(x)=1+[2cos^2(x/2)-1]=2cos^2(x/2). The integrand has a known antiderivative and the integral yields 1/2tan(x/2)+C

[@XphoneX502]

yeah! thats i also got the same answer......

[@khaledchatah3425]

1 thing the 1/(1+cosx) we can write it as 1/2(cos((1/2)x))^2 which is 1/2 sec^2(1/2 x) which is tanx/2 +c

[@antho8070]

This is one of the best KZbin video I've seen so far ! Thanks a lot !

[@NurHadi-qf9kl]

|.xx dx/¥(1-xx)=(ax+b)¥(1-xx)+C|dx/(1-xx) Permisalan tsb dideferensialkan untuk menemukan besarnya konstanta q&b Dan suku terahur kan menghasilkan arcsin

[@mintusaren895]

Integral and differential Depend on plus and minus . Apply zero or infinity.

[@mohammadmunzurulhaque3197]

Thanks a lot dear brother. Want to meet you in person and say " Thank you "

[@blackpenredpen]

Awww thank you. It’s my pleasure to help.

[@ellaevensen6132]

This is the greatest thing to ever have been produced.

[@thekingdragon6660]

Man those videos are some gold

[@ayeshafaisal7949]

You are doing a great service for the world. Much respect sir

[@random-playz3952]

Great video, loved the doremon intro as well

[@balakumarank2006]

The last integral can also be done by adding and subtracting the numerator by 1 and multiplying the entire term by - ve . It's quite easy..

[@audrey9903]

sir, you have single-handedly saved my GPA

[@jenitodani4894]

In que 10 In the place of 1+cosx = 2cos^2(x/2) Therefore 1/ 1+cosx = sec^2(x/2)/2 Which results in tan(x/2)/2+C 😊

[@NasirKhan-lq5jl]

YAY!!! I downloaded your worksheet just 1 hour ago i got em' all. And for the first integral we could also let u=ln(x) that can work too. Anyways KEEP UP THE GOOD WORK because this channel ROCKS....!!!!

[@blackpenredpen]

Yay!!! True, however, we still need IBP for the first one, so we might well do it right away : )

[@NasirKhan-lq5jl]

@@blackpenredpen yep.

[@kaiofkai]

Cleared a lot of concepts in just few minutes ,👍🏻 appreciated bro

[@nounoukasber]

I noticed that the 12th integral can be done with DI method I don't know if it's easier but you can recognise 1/sqrt(1-X^2) to be the derivative of sin(x) and you can D X^2 maybe this will be hopeful to some of you

[@inanis6707]

yes, this is a more sophisticated way, but is also really easy. It's just as simple as trig substitution with the only thing being that you have to use DI twice. also also, taking u = sin^-1x is the same thing as x = sinu.

[@golammartuzahossain6748]

the first integral can also be done by taking u=1/x, break the 1/x^3=1/x^2 x 1/x then the rest follows :3

[@elenadelgadoEHS1]

Hi, I love your videos. Ii used to doubt my ability to do integrals but you make it so easy. Thank you for sharing your knowledge and techniques. You are an amazing person to do this vlogs.

[@bournitolul8850]

At the 10th integral , we can use trig formula where 1+cosx=2cos²(x/2) which is then equal to 1/2 *(1+tan²(x/2)) and yeah integrating it is eash now.

[@IoTLearner]

Q10 can be solved more easily by simply replacing (1 + cosx)= 2(cosx/2)^2 as 1+cos2x = (2cosx)^2

[@abdallababikir9154]

no one knows that identity...

[@abdallababikir9154]

nvm ik it, 1/2(1+cos2x) is normal

[@shawnli9258]

Yeah I use that when I first came across this Q, it works out to be tan(x/2). Using his method it works out as cscx - ctgx = (1-cosx)/sinx. And guess what, we just prove the identify of tan (x/2)= (1-cosx)/2 using integral lol

[@K20-mbvn]

honestly for even sec^3x we can write it as (1+tan^x)^1/2 . sec ^x taking u =tan x no need for integration by parts when u mentioned it for even 2nd integral

[@omaraladib2165]

في التمرين الاخير عند جذر ال مقام وبما انه مربع يجب وضع القيمة المطلقة للنسبة المثلثية😊

[@fjworkshop]

please put part 2.this video is really helpful

[@CleanLAbeach]

Youre the best calculus professsor ever

[@sjf96]

At 21:00, "You did the hard work, so PUT IT DOWN." Great video showing integral strategies, thanks. Integral calculus has really shown me that it's the convergence of all the mathematics I've learned up this point - which is really cool

[@lol.hafsaa]

Everyone talking about calc 2. We're doing this in calc 1😭. Thanks for the video!!! It helped a lot

[@nolanpestano1798]

i cant describe how much i love you;

[@Ash_0021]

Well very good video its helping me in my high school calculus love from india