Elementary vs. Non-Elementary integral battles! (beyond regular calculus)

[@blackpenredpen]

Battle 1, integral of cos(x^2) vs integral of cos(ln(x)), @1:00 Battle 2, integral of ln(1-x^2) vs integral of ln(1-e^x), @7:55 Battle 3, integral of x^(x/ln(x)) vs integral of x^x, @16:23 Battle 4, integral of x*sqrt(x^3+4) vs integral of x*sqrt(x^4+4), @19:29 Battle 5, integral of x/ln(x) vs integral of ln(x)/x, @32:25 Battle 6, integral of ln(ln(x)) vs integral of sqrt(x*sqrt(x)), @34:00 Battle 7, integral of sqrt(sin(x)) vs integral of sin(sqrt(x)), @36:13 Battle 8, integral of sqrt(tan(x)) vs integral of tan(sqrt(x)), @40:52 Battle 9, integral of tan^-1(x) vs integral of sin^-1(x)/cos^-1(x), @59:13 Battle 10, integral of 1/(1-x^2)^(2/3) vs integral of 1/(1-x^2)^(3/2), @1:04:23 file: docs.wixstatic.com/ugd/287ba5_3f60c34605f1494498f02a83c2e62b29.pdf

[@chirayu_jain]

New challange for me😊

[@VibingMath]

wow nice timestamp! Should be pinned yrself!

[@yaleng4597]

Where are those special functions?

[@angelmendez-rivera351]

Yale NG Which ones are you talking about? They never appeared in the video.

[@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

[@hunter6549]

Another approach to the integral of ln(1-x^2) dx would be to factor the inside and then use the product rule of logarithms to get the integral of ln(1-x) + ln(1+x) dx. It's a bit easier to solve this way.

[@The1RandomFool]

Just a real minor point of #4: you could also do a hyperbolic trig substitution instead, and you'd get a simple inverse hyperbolic sine term in the final answer instead of the natural logarithm. That natural logarithm is also convertible to the inverse hyperbolic sine.

[@benjaminbrady2385]

Solution to integral of sqrt(tan(x)): There's a blackpenredpen video on that + c

[@helloitsme7553]

The way I like to think about the Integral of cos(x^2): with some clever substitutions and Euler's formula it can be shown that it can be written in terms of the integral of e^(x^2) and since that cannot be defined in terms of elementary functions, thus the integral of cos(x^2) cannot be

[@aashsyed1277]

heloo

[@holyshit922]

22:21 Euler's substitution sqrt(u^2+4)=t-u would be better idea here Last one third Euler substution (with roots) or integrating by parts also are good option

[@angelmendez-rivera351]

To integrate arcsin(x)/arccos(x) from x = -1 to x = t < 1, let x = cos(θ). Then dx = -sin(θ) dθ. The integrand is now -arcsin(cos(θ))·sin(θ)/θ. The bounds are from θ = π to θ = arccos(t). On the interval (0, π), which is the codomain and range of arccos(t), arcsin(cos(θ)) = π/2 - θ. Therefore, the integrand is -(π/2 - θ)·sin(θ)/θ. Factoring -1 will change the bounds to run from θ = arccos(t) to θ = π, with integrand (π/2 - θ)·sin(θ)/θ. By linearity, this gives the integrals of (π/2)·sin(θ)/θ and -sin(θ). The first integral is equal to (π/2)·(Si(π) - Si(arccos(t))), and the second is equal to cos(π) - cos(arccos(t)) = -(1 + t). Then the total integral is simply equal to [(π/2)·Si(π) - 1] - (t + Si[arccos(t)]). Call (π/2)·Si(π) - 1 = C, so the integral is simply C - t - Si(arccos(t)). Done! For the record, Si(x) is defined as the integral from s = 0 to s = x of sin(s)/s. We can extend the answer to other intervals, but this requires some caution, since arcsin(cos(θ)) = π/2 - θ is no longer true in other intervals.

[@GSHAPIROY]

15:05 In the last two terms of that answer (before the +C) it was not necessary to use absolute value around the ln input. Respond to this comment if you can figure out why!

[@iabervon]

On the first one, it was obvious, because cos(ln x)=(x^i+x^-i)/2. Power rule, separate real and imaginary coefficients, and put it back to trig functions. Even if you're not going to use complex numbers, you can guess the right integral because cos is like an exponential and goes well with ln and poorly with x^2.

[@alejrandom6592]

19:57 you can do both u-sub and trig-sub at the same time by letting x^2=2tan(theta) ;) then, xdx is nicely equal to sec^2 and the rest is just the usual

[@OOTMI]

I love your enthusiasm!

[@chirayu_jain]

I want to know, how to prove that the integral of a function is not elementary, please tell

[@blackpenredpen]

Chirayu Jain It’s quite hard to prove it mathematically. I think we need to know Galois theory from advanced abstract algebra in order to do so. I actually don’t have experience in it unfortunately.

[@chirayu_jain]

@@blackpenredpen, what a coincidence I started learning abstract algebra just 2 weeks before., 😁

[@japotillor]

Galios Theory, it's probably easier to just know which ones are non-elementary, rather than to prove each one individually.

[@angelmendez-rivera351]

Chirayu Jain You can prove the non-elementariness of an integral using the Risch algorithm.

[@jongyon7192p]

@@japotillor That by itself doesn't disprove that there might be some weird unknown way to do an integral.

[@ayushk3870]

Integration of e^-xx from +inf To -inf with pler co-ordinates

[@charlietlo4228]

20:00 you can directly let x = √(2tan(theta))

[@giovanni1946]

So nice to see a notification from bprp just after the first day of school :D

[@blackpenredpen]

Thanks!!!!

[@Armbrust666]

The second one was a bit over the top, ln(1-x^2)=ln((1-x)(1+x))=ln(1-x)+ln(1+x)

[@GhostyOcean]

Either way you need to do integration by parts. Personally, I broke up the ln but if makes sense to use IBP with a bit of work extra then go for it. As long as you get an answer and understand the process

[@-james-8343]

GhostyOcean no you don’t need to do integration by parts with the method he stated. After you split the ln you can split the integral and solve them both by u sub

[@angelmendez-rivera351]

-James- Integrating ln(u) requires integration by parts, so you are wrong.

[@GhostyOcean]

@@-james-8343 in order to integrate ln(x) you need to do IBP unless you have the answer memorized (xln(x)-x)

[@MG-hi9sh]

Gábor Tóth Tbh, it’s just as hard if you split it. I split it, and if anything, that made it harder because you have to do IBP twice.

[@bodor3139]

Take my love for this channel from Bangladesh.

[@wenhanzhou5826]

who else got a smile on the face at 16:15 because you have watched an old bprp video?

[@williamadams137]

Sun and clouds me

[@MG-hi9sh]

Sun and clouds Nah, I still messed it up, ffs. 😂😂😂

[@sinosodialajay797]

You are a great teacher

[@mokouf3]

Battle 2: Don't use partial fraction! Use ln(ab) = lna + lnb rule first, much more simple!

[@mcwulf25]

That was my thought. ln(1+x) + ln(1-x)

[@dottemar6597]

That's what I did - got two standard ones.

[@Mario_Altare]

I love these videos! Encore, encore :-)

[@seeeeeelf]

7:55 wouldn't that be easier to just factor 1-x^2 as (1-x)(1+x) and then use the log propertry to split the ln of the product?

[@Pageleplays]

15:15 „Integrale für Euch“ 😂 Grüße an alle Deutsche 🇩🇪🙌🏽

[@blackpenredpen]

SGE 1899 Hahahah yea!!! Lars helped me to translate it. : )

[@attamirza2602]

hahahah Ehrenmann

[@nchoosekmath]

Correct me if I am wrong, but at 8:50, you can factor 1-x^2 and use rule of log to expand it into 2 terms?

[@blackpenredpen]

Oh yes. Then integration by parts after that. Both work

[@nchoosekmath]

@@blackpenredpen Right, unless one memorize that integral of ln(x) is xln(x)-x hehe

[@blackpenredpen]

n choose k yea

[@KazACWizard]

integrating arcsinx/arccosx is actually doable;much easier to do than the other ones mentioned as undoable previously. its just a bit of subs and ibp and using the Si function.

[@byronrobbins8834]

We presently scratch the integral, if it is a non-elementary integral.

[@VibingMath]

One-hour long video but u definitely spent a lot more time than that! Your effort should be appreciated! And also the patreon list grows longer everytime 😁👍 PS it's 1am here in HK and yr thumbnail looks cool with some chill 😆

[@blackpenredpen]

Mak Vinci lollll thank you!! I prob will make another thumbnail tho. I don’t think that is that appealing lol

[@VibingMath]

@@blackpenredpen Hey keep this kind of thumbnail man(but not too many), it makes others curious to press the thumbnail 😁

[@thomasborgsmidt9801]

This is the best video You have made - of those I've seen. I was especially happy to know that ln(ln(x)) is a non-fundamental function. That question has been bothering me for years.

[@rbradhill]

one take, with some cuts. i dig it 😁

[@reu.mathematicsacademy8566]

Brilliant sir

[@jayapandey2541]

In India we have National Teachers' Day on 5th Sept. So, Happy Teachers' Day to BPRP and all other teachers in advance.

[@JamesLewis2]

You probably made that future video already, but it is interesting to point out that the most obvious attempt to antidifferentiate arcsin(x)/arccos(x) with respect to x results in the sine integral: A basic trigonometric identity has arcsin(x)=π/2−arccos(x), from which the integrand becomes ½π/arccos(x)−1; then the substitution x=cos(y) with dx=−sin(y)dy results in the sine integral. That is, ∫arcsin(x)/arccos(x) dx = -x−½π∫sin(y)/y dy = −x−½πSi(arccos(x))+C.

[@Dalton1294]

Here's another way to write the answer to question 2, xln(1-x^2)-2x+2tanh^-1(x)+C

[@not_intelligent5733]

√tanx i love this integral same as 1/(x^6+1)

[@robertl.crawford4369]

Lets see those special functions!

[@accountfantoccio5608]

Would it actually be faster to integrate cos(ln(x)) by using the complex definition of the cosine? You would then need to integrate (x^i+x^-i)/2, which is just a matter of integrating polinomials.

[@angelmendez-rivera351]

Account Fantoccio Relatively, yes.

[@adityakumarvishwakarma7282]

Sir please make a video on ramanujan formula on finding value of pi

[@chirayu_jain]

Oon Han has made a video on it

[@saradehimi4791]

Big salutation from Algeria thank you Allah blesses you

[@nuklearboysymbiote]

Number 8 was crazy

[@ishanbanjara734]

I came here after the rap battle in 8 Miles😂... I am ready for the battle!!!

[@حوداروك]

12:30 you could just directly integrate it to 2tanh^-1(x). instead of partial fractions.

[@angelmendez-rivera351]

حودا روك No, because the domain would be incorrect.

[@sinosodialajay797]

Please make a collaboration video with 3blue1brown together

[@adamzeggai5506]

YES

[@azujy2959]

YES

[@adamzeggai5506]

@@azujy2959 gosh that would be so cool

[@xxgoku7774]

Thumbnails are getting stronger

[@indrarajgocher7465]

Best videos sir for maths

[@EduardoViruenaSilva]

Second round: integral 1 / (1-x^2) = arctanh x + C

[@kingarth0r]

which integrals are intermediate and high school?

[@benjaminbrady2385]

Now solve the special function ones!

[@luizkemo]

What about x^dx? Can u do ir pls?

[@angelmendez-rivera351]

kemosabe What is that?

[@h.m.6228]

May the chenlu be with your integrals.

[@GSHAPIROY]

26:25 100 Integrals #61.

[@andrewwang164]

integrating ln(cos x) would be an interesting one

[@Ri_F]

the ad I had for this just said "Find your Steve" 😱😱😱

[@blackpenredpen]

!!!

[@seroujghazarian6343]

11:22-11:25 the integral of the thing you are saying needs partial fractions doesn't, actually, because the answer is clearly inverse hyperbolic tangent (Argthx/Argtanhx)

[@angelmendez-rivera351]

Serouj Ghazarian Well, that's not correct either, since the domain or arctangent is different from the domain of the function we started with. Strictly speaking, partial fractions are the only correct way to get the most general antiderivative, and this can be proven.

[@seroujghazarian6343]

@@angelmendez-rivera351 ArGtanH, not arctan

[@seroujghazarian6343]

@@angelmendez-rivera351 the function we started with is ln(1-x^2), which has EXACTLY the same domain as Argtanh.

[@jamez6398]

My god, integral of x times the square root of (x^4 + x) is a really complicated integral. It would be even more complicated if one had to integrate sec^3(x) from scratch... 34:26 The integral of the square root of (x times the square root of x)?? The integral of the square root of (x + the square root of x)... 🙂 The integral of √(x + √x) Or the integral of 1/√(x + √x) Or the integral of 1/√(1 + √x)

[@angelmendez-rivera351]

James Oldfield Obviously, it is sqrt(x·sqrt(x)). Also, the integral of x·sqrt(x^4 + x) is non-elementary, and is also not the integral dealt with in the video, and the one in the video was actually very simple.

[@angelmendez-rivera351]

Also, integrating sec(x)^3 from scratch is fairly easy too.

[@jamez6398]

@@angelmendez-rivera351 You must be a really smart person to find this kind of thing easy. I'm still at the level of basic integration and differentiation, power rule stuff. Like 1/cube root (9x^4) + 3x^3 + x^2. Really, really basic stuff like that...

[@jamez6398]

@@angelmendez-rivera351 I was being cheeky. I know he said √(x√x). I was thinking it was easy (relatively), and that √(x + √x) would be a harder integral to do...

[@angelmendez-rivera351]

James Oldfield I wouldn't say I'm smart, just math savvy. Anyway, I only said it's easy because that was one of the easier integrals showed in the video. Most of the other ones were more complicated. And it doesn't have anything on the integral of sqrt[tan(x)], or even worse, the cbrt[tan(x)] integral. The integral of sqrt(x + sqrt(x)) is indeed more complicated than the integral of sqrt(x·sqrt(x)). In fact, the integral is very clever. For example, if y = x + sqrt(x), then dy = [1 + 1/{2·sqrt(x)}]dx. Thus, sqrt(x + sqrt(x)) = sqrt(x + sqrt(x))[1 + 1/{2·sqrt(x)}] - sqrt(x + sqrt(x)/{2·sqrt(x)} = sqrt(x + sqrt(x))[1 + 1/{2·sqrt(x)}] - sqrt(sqrt(x) + 1)/2. Now one can split the integral in two parts using linearity. The integral of sqrt(x + sqrt(x))[1 + 1/{2·sqrt(x)}] can be found using the very simple substitution I already mentioned, and this integral will be equal to (2/3)·sqrt(x + sqrt(x))^3 + C. All the remains is evaluating the integral of sqrt(sqrt(x) + 1). Let z = sqrt(x) + 1, so x = (z - 1)^2, and dx = 2(z - 1)dz. This leaves the integral of 2z^(3/2) - 2z^(1/2) with respect to z. This is just a very basic power rule integral, and it gives the antiderivative (4/5)z^(5/2) - (4/3)z^(3/2) + C. Substitute back to get (4/5)·sqrt(sqrt(x) + 1)^5 - (4/3)·sqrt(sqrt(x) + 1)^3 + C. Altogether, the integral of sqrt(x + sqrt(x)) is nicely equal to (2/3)·sqrt(x + sqrt(x))^3 + (2/3)·sqrt(1 + sqrt(x))^3 - (2/5)·sqrt(1 + sqrt(x))^5 + C.

[@jeunefofanaadamadelecolede7659]

salut monsieur svp j'aimerais avoir un pdf des 100 integrale ou un pdf d'çntegrale pour licence de mathematiues svp

[@kaandogan2470]

Hey BPRP , can you make a video about Group Theory ?

[@ssdd9911]

can show hyperbolic functions more love or not?

[@Промо-в1ю]

It will be a great pleasure to me, if you explain how to separate elementary from nonelementary ones. Does such formular exist?

[@angelmendez-rivera351]

Промо Risch algorithm.

[@centugurdag7776]

Hi, cos(X square) is a function . Geogebra gives a result, if you integrate ( calculate the area) between 2 points Why we can say that this integral does not have a result.thank you For your reply

[@angelmendez-rivera351]

Cent Uğurdağ Because the antiderivative of cos(x^2) is *not* the area. The antiderivative of cos(x^2) is simply another function, but the area under the curve is a number. Not remotely the same thing. Any software can calculate any area, but if you ask Geogebra to give you the antiderivative, it *cannot* and *will not* give you an answer, because there is no answer.

[@centugurdag7776]

İ agree but want to know why there is no antiderivative of this function

[@tjli7472]

Hey Im a Calculus amateur. Just wondering what method did bprp used at 38:50. Thx in advance!

[@CruzW123]

Hi! Four years later, are you still a calculus amateur?

[@VaradMahashabde]

Question 3, the absolute troll

[@juanjoselezanomartinez5714]

Good video, can you please help me with this integral .. X*Sec(X)

[@not_intelligent5733]

Integration by parts X take D and I sec x Integration of secx is log|secx + tan x| and then its easy

[@justabunga1]

It's non-elementary because if you try to do IBP, you get xln(abs(sec(x)+tan(x)))-integral of ln(abs(sec(x)+tan(x)))dx. Here integral of ln(abs(sec(x)+tan(x))) is non-elementary.

[@Anders3000]

What font did you use in your document? Do you use LaTeX package or?

[@falkinable]

For #9, the ln part turned out to be ln|cos(arctan(x))|, anyone else have this??

[@angelmendez-rivera351]

Freddie Correct

[@mikedavis7636]

Isn't it instead of using partial fractions, Can we not have xln (1-x²) -2x + tanh-¹ (x) +c ? As the answer?

[@borntofight5887]

Can you solve it Int. (x-2)/[(x-2)^2(x+3)^7]^1/3

[@chirayu_jain]

How do you make your thumbnail🙏😊

[@blackpenredpen]

I use “page” on Mac, math type and pictures.

[@chirayu_jain]

blackpenredpen can you please give any suggestions for android phone or windows laptop as we don’t have an MacBooks or IPhones or iPads with us.

[@anhadrajkhowa5850]

Yall I was just vibing to the Doraemon theme song in the beginning.

[@nchoosekmath]

58:05 is just insane lol

[@blackpenredpen]

n choose k yea! And I didn’t do partial fractions just to save time. Lol

[@halaalp9706]

Why IS integral of tan (sqrt x ) impossible to solve I genuinely don't understand

[@moon-ia2068]

can you know if the integration is possible or not just by looking at it ? , and if yes how do you know ?

[@krabbediem]

Hi BPRP, and thank you for the videos :D I guess this comment will go unnoticed, but if I never ask, I'll never know :) Why are half of these functions impossible to integrate? You just mention as a fact that it's impossible but never why. I'm not great at integration, so I don't understand _why_

[@jmadratz]

Do you think that Isaac newton would have been able to derive all of these integral solutions back in his day

[@cyruscyros1891]

On question number (8). Suppose you let integral equal to Q, then square both sides and integrate twice then take the sqr,, can it work?

[@rurafs7934]

Wait... 1 hour 😯💚

[@muscleeagle_]

I never forget the chendu😆

[@Lamiranta]

bprp: *showing 8 integral battle* me: ...here we go again

[@angelmendez-rivera351]

Konstantin Cherkai 10*

[@herlysqr1650]

How we can know what is elementary and what is not?

[@vijayrathore4811]

Sir ,What is the integral of ∫(1-x^2)^n dx

[@Proximachannel]

I like your microphone

[@jarogniewborkowski5284]

Did You make already any video with non-elementary integrals like eliptic ones?

[@oscartroncoso2585]

LETS GO!

[@sinosodialajay797]

On 14 September it is teacher's day in India . Please make a excellent special video on the day.

[@Paul-ob2hy]

for number 2, isn’t the int of 2/1-x^2 just 2arccot(x)?

[@angelmendez-rivera351]

Yes.

[@asparkdeity8717]

It’s 2artanh(x), like the hyperbolic inverse tanh function

[@bryangohmppac6417]

Sir, why don't you make a video about proving that the ramanujan formula

[@jackhounsom8867]

Isn’t it easier on the 2nd one to change it from ln(1-x^2) to ln((1-x)(1+x))=ln(1-x) + ln(1+x) and integrate like that?

[@angelmendez-rivera351]

Jack Hounsom Eh... it's about as easy, but it depends

[@MG-hi9sh]

Jack Hounsom Nah, it’s worse, I did it, and trust me, it’s worse.

[@mathswithpana]

hello brother. I get a different answer for number 2 intergral ln(1-x^2)dx instead of 1-x i get x-1 and 1+x is same as x+1

[@urvpatel829]

BPRP how to find range of Sinx-√3.cosx+1

[@iabervon]

There's a formula for turning a linear combination of sin and cos into a single sin (or cos) with a phase shift and coefficient. Then you just need to adjust the range for adding 1.

[@MikFrost00]

But how do we know it's impossible to integrate cos(x^2)?

[@MikFrost00]

@@1315-z1p yh that is true but how do we know that

[@MikFrost00]

@@1315-z1p hmm I think there is a proof in Galois theory

[@mohammadzuhairkhan8661]

For no. 8, can't we split 1/(t^2-2) into partial fractions and use ln? It is much friendlier than coth. Also, why coth instead of tanh?

[@blackpenredpen]

Yes. But it would be just longer...

[@mohammadzuhairkhan8661]

@@blackpenredpen But why coth instead of say tanh? According to you they are identical...

[@angelmendez-rivera351]

Mohammad Zuhair Khan ln in this situation is not friendlier than ln, since the inside of ln would be a complicated expression. In fact, coth is expressible in terms of ln, so that makes your point moot.

[@MG-hi9sh]

blackpenredpen Tbf, I prefer it because you can see how you get the answer, whereas the tanh is just a standard formula.

[@dkravitz78]

Number 2 way easier to write ln(1-x^2)=ln(1+x)+ln(1-x)

[@Ni999]

I hate that solution for ∫ √tanx dx and prefer this one - ∫ (√tanx + √cotx)/2 dx + ∫ (√tanx - √cotx)/2 dx Use the common denominator, √sinxcosx and split 2 into √2*√2, and rearrange - √½∫ (sinx+cosx)/√(2sinxcosx) dx + √½∫ (sinx-cosx)/√(2sinxcosx) dx Wouldn't it be nice if we had a way to use sin²x + cos²x = *1* on the bottom? *2sinxcosx* = 1 - ( *1* - 2sinxcosx) *= 1 - (sinx - cosx)²* = ( *1* + 2sinxcosx) - 1 *= (sinx + cosx)² - 1* Substitute each one - √½∫ (sinx+cosx)/√(1-(sinx-cosx)²) dx + √½∫ (sinx-cosx)/√((sinx+cosx)²-1) dx Substitute t = sinx - cosx u = sinx + cosx √½∫ 1/√(1-t²) dt *-* √½∫ 1/(√u²-1) du = √½sin¯¹(t) - √½cosh¯¹(u) + C *Substitute back for t and u and you're done,* unless you prefer to use ln|u+√(u²-1)| in place of cosh¯¹(u). If so remember that √(u²-1) = √(2sinxcosx) and everyone converts that to √sin(2x) for no useful reason I can see but there it is if you want it. Note that √½ is really (1/√2) but I don't have all day to type that and you don't have all day parsing parentheses in a KZbin comment. I'm sure it doesn't matter and it's probably just me but I find that solution a whole lot cleaner, easier to follow, and easier to remember with fewer chances of making an algebra mistake. The long, drawn out version is called _Trigonometric Twins_ (not my video) at kzbin.info/www/bejne/moWbk2utds-Hga8 and watch out for the typo near the end. You probably need to learn the method bprp showed to pass a test though. I don't know. I also find the similarity of the intermediate form compared to the algebraic answer in the video pretty interesting. √½∫ (√tanx + √cotx)/√2 dx + √½∫ (√tanx - √cotx)/√2 dx

[@angelmendez-rivera351]

Ni999 Wow, well, this is just extremely pedantic as a comment. Let me address a few things: 1. This is not that much simpler to what is on the video, contrary to what you claim. And the answer he gave was not in its simplest terms, so disputing elegance there is futile. 2. 2sin(x)cos(x) is simplified to sin(2x) because it is, well, *simpler.* Individual trigonometric functions are always preferable to products thereof. 3. BPRP's method is generalizable to higher order roots of tan(x), whereas yours is not. And considering the precedent this has on the channel, it makes perfect sense he explained it the way he did it.

[@Ni999]

@@angelmendez-rivera351 Ok. 1. It's simpler *for me.* At the end of his solution, bprp had to look more than once to make sure of the substitutions at the end. I hit that same thing every time using the algebraic method for this particular problem. If you say it's not more elegant, fine. It's certainly easier *for me* to finish the substitutions. 2. If I'm using it to solve for a definite integral, and I've already pulled up (or coded) the solutions for sinx and cosx and stored them, then it's easier to multiply the two stored values than to pull up a third trig function. 3. Bprp has a video showing 4 ways to solve ∫ secx dx. The other three ways provide beneficial exercise and food for thought. He even has an alternative video (can't remember it off hand) where he shows integration using this same method - ∫ f(x) dx = ½∫ f(x)+g(x) dx + ½∫ f(x)-g(x) dx. So it's not just "my way" and he didn't avoid showing the overall method elsewhere because it couldn't be generalized. I never said that one ought not learn what he taught. I even said that you'd probably need to know his way for a test - that instrument to exhibit long-term learning. I thought that others who had missed the method would find it interesting. I'm not going to apologize because you found my comment pendantic - especially given that you felt the need to resort to numbered paragraphs. I thought this was also math for fun and anyone is free to agree and laugh with me or disagree and laugh at me. Either way, it's all good. Clearly I was dead wrong. Let me know if you want me to delete the comment (and therefore the thread), it makes no difference to me. Everyone who knows this channel knows who you are and respects you. I won't be bothering you again.

[@rurafs7934]

How to do that (long division)?

[@pradoalan6818]

56:36

[@surajsanganbhatla5905]

Only between you and me!😁

[@felixangelsanchezmendez1466]

Could you solve this integral? Integral of (secx)^(3/2). I wish you did it. Thanks for giving a lot of support

[@angelmendez-rivera351]

Angel Mendes This integral is non-elementary, so there is no solution anyone can give you that you would be satisfied with.

[@felixangelsanchezmendez1466]

Thank you so much, bro

[@ayushjuvekar]

Hey bprp, what font do you use in your files and thumbnails?

[@SR-kd4wi]

Can you teach us group theory?

[@benjaminbrady2385]

Lol, I speak Irish but I don't know if that helps in the slightest

[@warrickdawes7900]

Almost an f-bomb at 27:35!

[@pichass9337]

What do you mean by elementary?

[@angelmendez-rivera351]

Liam Martin 2: Electric Boogaloo Cannot be expressed as an algebraic combination of polynomials, exponentials, factorials, ordinary trigonometric or hyperbolic functions, absolute values, or the inverses of any of the above.