thanks , the integral is really amazing. so intelligent. Mathematics is deeply bottomless, and probability and statistics are more difficult to understand. Gamma distribution, chi-square distribution, is so hard to understand.
This is probably the most fun I've ever had watching a video about integrals! It was a long journey, but it was worth every second. Thankyou for such a great video!
@@blackpenredpen But why on earth would you do that second substitution at all? Isn't there another way maybe more intuitive..like integration by parts Ibwas thinking..Hope you can respond when you can.
when you say "please do not ask me to check the answer by diferentiation", I'm pretty sure you were really thinking "please ask me to check the answer by diferentiation" ... So, in order to please you..... Would you check the answer by diferentiation???? Please :)
scrolled down expecting the comment "check the answer by differentiation" as top or only comment .. urs was 3rd and has a bunch of text around it internet, you disappoint...... (altho the expectation ur not meeting, is actually... so in a way.... yay! ....)
I was wondering this myself lol at the beginning of this video. I'm betting it inevitably leads to just letting u= the n-th root of tan(x) and then integrating the resulting rational function using partial fraction decomposition - maybe synthetic division for powers of tangent not = 1 or 2.
This should be easy. Separate tan^n into tan^(n-2) and tan^2 then integrate by parts. Then substitute sec^2-1 for tan^2 and you get a reduction formula. Evaluate integral of tan^2 and that should be enough I guess.(I havent tried this but I think this works for even exponents only.)
16:56 Yeah, putting down some TNT on that would definitely help :) 18:18 It's quite interesting that `1/2` and `√3/2` appear in the completed square, because: a) they are the real and imaginary part of the cube root of `-1` that appear in the complex factorization of the denominator; b) they are the sine and cosine of the 60° angle at which this cube root lays with respect to the real axis :) c) we've got the cube root (of the tangent) in the original integrand. 21:30 We have to go deeper... :> 23:52 The dream is collapsing :J
Evolution can you give me more letters for my substitions into different integral worlds? Sure blackpenredpen ACTUALLY USES OLD LETTERS FOR SUBSTITUTIONS LIKE A BOSS
The second integral can be done without any substitution: Multiply and divide by 2 (you can factor out the divide by 2 later, with the final constant distribution): (2t+2)/(t²-t+1) The derivative of the denominator is 2t-1. So, write the integral as: (2t-1+3)/(t²-t+1) Split the fraction: (2t-1)/(t²-t+1) + 3/(t²-t+1) The first one's integral is ln|t²-t+1|. For the second one, complete the square as (t-1/2)²+(√3/2)². The integral is 2√3*invtan((2t-1)/√3). Now, distribute the divide by 2: 1/2*ln|t²-t+1|+√3*invtan((2t-1)/√3).
This couldn't have been uploaded at any time better, I'm just about to fall asleep and I always trance at your videos which makes me go to sleep! Thank you so much! :D
I thought he will forget the + C part Truly professional to the core I am starting to fall in love with math and pens And of course our one and only *CALCULUS*🥳🥳
i never´ ve seen a intgral like this . i been watching all your videos and in my opinion is very useful. in other side i want to thank, you videos are helping me to improve my english how you know ,y native language is spanish, but thanks for share it. congrats
This is way late, but you can further compactify the answer (and show off some extra algebra skills) by taking advantage of log properties like so: 1/2ln[ sqrt( tan^4/3 x - tan^2/3 x + 1 ) / ( tan^2/3 x + 1 ) ] + sqrt(3) arctan( (2 tan^2/3 x - 1)/sqrt(3) ) + C.
i just love your videos man... i just want to say that you helped me overcome my fear of integrals.. and your approach to not just this video but all the others are just brilliant.. thank you.
I kept on making mistakes when trying to check this answer by differentiation, and had pledged to keep this video tab open until I had successfully checked it. So pleased to say that I finally got it right today! I can close the tab now. :D
This channel is now offically a MATH MEME ! "How about the integral of [fancy variation of sqrt(tan x)] ?" - bprp, you have a long meme carrier ahead...
if they´re not nice they get this integral on their exam! how about that? or worse sin(x)/x from 0 to infinity (but don´t ask for the steps. that´s overkill. though maybe you can show us that one? i didn´t see a single video on YT about that ntegral i would have understood. and the only idea to even make this possible would be using the mclaurin series of the sine)
Just substitute the i's in the right places! tan(x) = -i tanh (ix), so you get tanh(x)=i tanh(-ix). Substitute this into the original integrand you get the cube root of i which is exp(i*pi/6), then you get an i outside when you replace dx with d(-ix). So the answer to integrating cube root of tanhx is just the original answer with all the x's replaced by -ix's, and an extra factor of exp(i*pi*2/3). Somehow you should be able to simplify everything in the end expression to show that the thing is real XD
So how does the monster grow in the complex domain? I would love to see the secrets in there so that I might apply them to quantum mechanics (and my cool design for a new warp drive model).
The interesting thing would be a professor taking the solution of that integral and making the class find the derivative maybe as a bonus problem on an exam.
I want to see this too, although I'll try it myself first because it may be possible that it be not expressible in terms of elementary functions or with a finite definite expression.
Here is a simpler and reliable way to do any partial fraction, t = A (t^2-t+1) + (Bt+C) x (t+1) = (A + B) t^2 + (-A + B + C) t + (A + C). Therefore, A + B = 0 = A + C and -A + B + C = 1. Therefore, B = -A = C = 1/3. This is instead of plugging random values.
Very thorough explanation. As an idea for another integral video, this is one of the most difficult integrals I ever got in college : Integral [ 2x^3 / ( ( x^6 + 3 ) ^ (1/2) ) + ( 3x^2 / (4 - x^2) )^(1/6) ] dx From x=0 to x=1 This could be the "scariest integral on RUclips" ;)
An excellent video, explained clearly. Really enjoyed watching the video and even had a try myself after watching. Great job. 我非常喜欢这个视频关于积分。黑笔红笔老师的解释清楚！看到了这个视频我自己试一试解决这个积分！
Small technical error: In the final function, you used the constant C to represent the constant of integration. BUT, you already used C to represent a different constant in your partial fraction decomposition. So, the final solution should use a different constant, like C-sub-1 or K.
Wolfram gives a general solution in terms of hypergeometric functions: int [tan(x)]^a dx = 2F1(1, (a+1)/2; (a+3)/2; -tan(x)^2) . ([tan(x)]^(a+1))/(a+1) + C In effect, a hypergeometric "correction" factor applied against a power rule-like construct operating on tan(x), as though you naively integrated it. Hence int [tan(x)]^(1/3) dx = 2F1(1, 2/3; 5/3; -tan(x)^2) . ([tan(x)]^(4/3))/(4/3) + C. which to me is a lot nicer. One of the things about maths is that there isn't one "true" or "best" _representation_ of a mathematical object, but unfortunately, the way it's taught conveys this idea that there is.
...And when you want to take a break from listening to this video, you are going to SUBSTITUTE IT for one of the easier/more beautiful ones to understand :D Like sin(z)=2
"and then we have to get back to the U world, and then we have to go back to the X world." What a wild ride - we're going on interplanetary journeys solving this integral!
OK, I checked the answer and it was correct! ruclips.net/video/mUXdgY5MeEg/видео.html
Happy New Year!!!
I approve.
Lol, just finished the vid and relaised this
Oh god my maths test is on 1st Aug and I haven't prepared for it yet...... integrals is also in test!😢😢😢
thanks , the integral is really amazing. so intelligent.
Mathematics is deeply bottomless, and probability and statistics are more difficult to understand.
Gamma distribution, chi-square distribution, is so hard to understand.
i really need your help to intigrate this for me i really need it ----- x-1*sqrt.2x-x*2
/x+1
Somewhere out there, a professor is gonna watch this video and make this problem a test question. RIP.
Sheev Palpatine Saw the comment, liked for the name
Sheev Palpatine I just had to pin this comment.
oh no, don't give them ideas
Sheev Palpatine that would be acceptable if that was an extra credit question where if you get it right you get full credit for the exam
Sheev Palpatine did you ever hear the tragedy of darth plagueis the wise?
Check the answer by differentiation
You mad man
You'll break him
I didn't understand the joke until I saw the answer XD
Avi Mehra The mad lad actually did it
Integrate it again.
You know it's a hard problem when he has to use 3 colors
BT7M yup
blackpenredpenbluepen
This is not hard prblm we indians take 30 sec to solve this
@@g.peloni3694 none cause we never and then we're like wtf are we doing.
@@g.peloni3694 uff...😂I'm also An Indian I'm a Calculus Lover too...
8:15 *erases that additional line*
ohh god thank youu
yoav carmel lollllll nice catch!!!!
Android Ninja yup
Wow
I don't want you to check your answer by differentiation. I want you to integrate your answer :D
BRUH
xD
You're a monster
That’s easy
@@asparkdeity8717 yeah ok
This is probably the most fun I've ever had watching a video about integrals! It was a long journey, but it was worth every second. Thankyou for such a great video!
Riley Wells my pleasure!!!!
@@blackpenredpen But why on earth would you do that second substitution at all? Isn't there another way maybe more intuitive..like integration by parts Ibwas thinking..Hope you can respond when you can.
I think you did a step wrong somewhere, please differentiate it to check.
Lolololol
pls no 😭
I love this. Great example of when to actually use partial fractions, too. I'm adding this channel to my list of favorite math channels! 🤓🤓🤓
Natalie Euley thank you Natalie!
You're going from blackpenredpen to 3blue1brown
How so? His vids are way cooler
Because this is really REALLY complicated yet you manage to explain it so well. Its cool at the same time
blackpenredpen Because of the color markers you are using
blackpenredpen and sometimes blue : )
blackpenredpenbluepengreenpen
when you say "please do not ask me to check the answer by diferentiation", I'm pretty sure you were really thinking "please ask me to check the answer by diferentiation" ... So, in order to please you..... Would you check the answer by diferentiation???? Please :)
lol! I am waiting for some viewers to do it!
blackpenredpen We want you to check it by differentiation. It's always easier than the integration, right?
blackpenredpen I proved it using my calculator. Is that good enough?
scrolled down expecting the comment "check the answer by differentiation" as top or only comment ..
urs was 3rd and has a bunch of text around it
internet, you disappoint......
(altho the expectation ur not meeting, is actually... so in a way.... yay! ....)
Maybe?!
I have a challenge for you. Make the generalization of the integral of n-th root of tan x dx
I was wondering this myself lol at the beginning of this video. I'm betting it inevitably leads to just letting u= the n-th root of tan(x) and then integrating the resulting rational function using partial fraction decomposition - maybe synthetic division for powers of tangent not = 1 or 2.
of course, I'm ignoring any possible trig manipulations ...
This should be easy. Separate tan^n into tan^(n-2) and tan^2 then integrate by parts. Then substitute sec^2-1 for tan^2 and you get a reduction formula. Evaluate integral of tan^2 and that should be enough I guess.(I havent tried this but I think this works for even exponents only.)
Kyro well Oh yeah..
Lol that's sadistic!
16:56 Yeah, putting down some TNT on that would definitely help :)
18:18 It's quite interesting that `1/2` and `√3/2` appear in the completed square, because:
a) they are the real and imaginary part of the cube root of `-1` that appear in the complex factorization of the denominator;
b) they are the sine and cosine of the 60° angle at which this cube root lays with respect to the real axis :)
c) we've got the cube root (of the tangent) in the original integrand.
21:30 We have to go deeper... :>
23:52 The dream is collapsing :J
Ah!!! I like the pt u made at 18:18... and what u said "The dream is collapsing"
I fear you may run out of substitute variables one day :P
Shakir Ahsan Romeo probably. Next I will use alpha, beta
*creates new alphabet after using all latin, greek, cyrillic, hebrew and arabic letters*
Evolution can you give me more letters for my substitions into different integral worlds?
Sure blackpenredpen
ACTUALLY USES OLD LETTERS FOR SUBSTITUTIONS LIKE A BOSS
After that start using ♤♡◇♧¤▪☆
How about हिन्दी
30:40 "Please, do not ask me to check the answer by differentiation"
Hey, you should totally check the answer by differentiation!
Now that is just incredible! By far one of the most difficult integrals I have ever seen and you have solved it so elegantly and so well.
Worth every second of the watch. Thank you!
200%productions thank you!!
This problem was soooooo easy! I solved it when I was a foetus. Hats off to this young gentleman realising the foetal power of solving integrals!
Fetus?
Bro really spelled it with an o 💀
@@JMZReview diarrhoea
Now do the integral of (tanx+(tanx)^(1/2)+(tanx)^(1/3)+(tanx)^(1/4)+(tanx)^(1/5)...)
Mr. Gentlezombie spoiler alert
this series will diverge, this question doesn't make sense.
@@aneeshsrinivas9088 D'oh!
The second integral can be done without any substitution:
Multiply and divide by 2 (you can factor out the divide by 2 later, with the final constant distribution):
(2t+2)/(t²-t+1)
The derivative of the denominator is 2t-1. So, write the integral as:
(2t-1+3)/(t²-t+1)
Split the fraction:
(2t-1)/(t²-t+1) + 3/(t²-t+1)
The first one's integral is ln|t²-t+1|. For the second one, complete the square as (t-1/2)²+(√3/2)². The integral is 2√3*invtan((2t-1)/√3).
Now, distribute the divide by 2:
1/2*ln|t²-t+1|+√3*invtan((2t-1)/√3).
Doctor: You have 31 minutes and eight seconds to live.
Me: immediately opens this video
This couldn't have been uploaded at any time better, I'm just about to fall asleep and I always trance at your videos which makes me go to sleep! Thank you so much! :D
I thought he will forget the + C part
Truly professional to the core
I am starting to fall in love with math and pens
And of course our one and only
*CALCULUS*🥳🥳
30:40 differentiASIAN
Lmao
Video: “Easiest integral on RUclips”
Me: oh that’s nice let’s check it out
Also video: equation is written very small and 31 minutes
Me: oh no...
Ok to the one dislike on the video I want you to post a follow up showing everyone how you do this integral.
Mace Jr there is no dislike now
NestorV S Hhh he couldn't do it.
Nah two people did.
disliking =/= "i can do better"
ask any sports fan if you don't believe me.
Now 5³ dislikes
Can you check the answer by differentiation ?
Pedro Bessa maybe one day :)
Really quick way to check. Takes 30 seconds. Plug in some values the the limit of integration and use fund. Thm of calc p2.
Could you check the answer by differentiation?
no
yes
Maybe
Fantastic job of explaining every step!
Best problem ever!
Tour de force for methods of integration.
I counted eight different tricks.
TI-84 plus showed it correct over:
-2π
Please differentiate this to get back to cuberoot of tan x
I love how enthusiastic you are! What great videos you make.
i never´ ve seen a intgral like this
. i been watching all your videos and in my opinion is very useful. in other side i want to thank, you videos are helping me to improve my english how you know ,y native language is spanish, but thanks for share it. congrats
Thank you so much. I am very happy to hear your nice comment! Keep up the good work too!
25:49 let's all appreciate how his only mistake in the video was using the wrong coloured marker
what about the tan(x) root of tan(x)?
I think impossible
NestorV S I thought you could integrate anything you wanted
Ace Eternus no, there are functions that their integral have no solution. A classic example of this is the function f(x)=e^x^2.
Ace Eternus
Some functions have no elementary anti derivative take x^x for example, you have to settle for a series.
Gregory House Tecnically there is a solution, but it just can't be expressed in terms of exponentials, polynomials, logarithms etc
This is way late, but you can further compactify the answer (and show off some extra algebra skills) by taking advantage of log properties like so:
1/2ln[ sqrt( tan^4/3 x - tan^2/3 x + 1 ) / ( tan^2/3 x + 1 ) ] + sqrt(3) arctan( (2 tan^2/3 x - 1)/sqrt(3) ) + C.
Great, now integrate (tanx)^1/5
I'm so happy to see a "BIG INTEGRALS NO SHORTCUTS" playlist, will watch them all
Imagine doing a test and solving this problem. You did every step right, but you forgot to write plus C
And professor deducts 1mark for that
@@delta2884 my professor would tear the whole thing and give you 0 ;)
Easier when in France they ask for a antiderivative so you don't have to add +C
Calc II + Calc II
i just love your videos man... i just want to say that you helped me overcome my fear of integrals.. and your approach to not just this video but all the others are just brilliant.. thank you.
I kept on making mistakes when trying to check this answer by differentiation, and had pledged to keep this video tab open until I had successfully checked it. So pleased to say that I finally got it right today! I can close the tab now. :D
BlackFiresong wow nice!!! I will check it one day too.
Thank you so much for replying! Love your work. :) Please do keep the crazy challenges coming. You have really helped to rekindle my love of Maths!
BlackFiresong I am very glad as well!! Thank you for watching my videos :)
Watching this made me very relaxed. You sir deserve a subscribed!
This channel is now offically a MATH MEME !
"How about the integral of [fancy variation of sqrt(tan x)] ?" - bprp, you have a long meme carrier ahead...
oscarjd74 OH WOW!!!! YOU ACTUALLY DID IT!
NEAT, instant screen saver ^^
How cool iz dat?! HAHAHAH
Jesus Christ! With this you have won a new follower! Subscribed!
Can you make some videos about the strategies of choosing the right integration technique for different problems?
Bon Bon will try.
Gratulálok, ez a szinte végig helyettesítéses módszer bemutatta, hol kezdődik az elméleti matematika !
Good. this is a high level problem in my sheet .
For B in 11:45, I like to multiplie by t in both side, and by going in the limit in +infini, we have 0=A+B
please check the answer by differentiation
very very good video and very satisfying to have a clean solution in the end.
This is so great! I wish I still had the "mathematical muscle" to do this myself. Maybe one of these days...
Hey man just checking in, its been 4 years hows that math muscle working out for you? I challenge you to do 3 integrals! You can do it
@@thelegendofme7520 Aaany day now... ;)
Glad you enjoyed yourself with that. This is a great review of a lot of techniques.
I'm worried for you.
MistaTwoJeffreyTwenty Yaay
U should worry about my students
blackpenredpen oh noooo, show them sympathy please, they don't deserve this!
Lol! I am teaching a calc2 class in the fall. We will see!! : )
blackpenredpen the next year i will see integrals on our tests and i need a bless from Pitagoras to pass these exams
if they´re not nice they get this integral on their exam! how about that? or worse sin(x)/x from 0 to infinity (but don´t ask for the steps. that´s overkill. though maybe you can show us that one? i didn´t see a single video on YT about that ntegral i would have understood. and the only idea to even make this possible would be using the mclaurin series of the sine)
you are such an amazing teacher you make this advanced stuff seem so easy
You earned yourself a new sub
Thanks sir .
Glad to hear! Thank you!
You know what you have to do now? Now you have to do the integral of (tanhx)^(1/3). These videos are the best please keep posting them!
Tom Himler Edward can help me with that. :)
Just substitute the i's in the right places! tan(x) = -i tanh (ix), so you get tanh(x)=i tanh(-ix). Substitute this into the original integrand you get the cube root of i which is exp(i*pi/6), then you get an i outside when you replace dx with d(-ix). So the answer to integrating cube root of tanhx is just the original answer with all the x's replaced by -ix's, and an extra factor of exp(i*pi*2/3). Somehow you should be able to simplify everything in the end expression to show that the thing is real XD
This does turn out to be a monster of an integral "doesn't it" :-)
Yea, isn't it?!
So how does the monster grow in the complex domain?
I would love to see the secrets in there so that I might apply them to quantum mechanics (and my cool design for a new warp drive model).
The interesting thing would be a professor taking the solution of that integral and making the class find the derivative maybe as a bonus problem on an exam.
Please make a video about the nth root of tanx
yes i agree
Yes, that'd be great! :)
I want to see that too
tan(nx) ^ (1/n)
I want to see this too, although I'll try it myself first because it may be possible that it be not expressible in terms of elementary functions or with a finite definite expression.
This is just what I needed for my summer.
I was kind of hoping to see you dancing in celebration chanting "One Take!" at the end
who knows maybe its 10th take
Notice that I didn't edit this video!
MrQuantum "one take" and "1st take" are different.
Mark Zero true. One take for this tho. I think it was my 3rd take bc my intros weren't good enough.
Mark Zero k
And my professor is expecting me to solve that in only 10 minutes.
: )))))
If every question on the test is like this Mr. Chow, I might have to clear my whole weekend to complete it.
Welcome to Calc2!
: )
here i am watching 30 minutes of math porn and not knowing wtf is going on
Clement Tan isn’t that the case when watching regular porn too?
Жиза
seems like a deep breathtaking moment when you got the answer to be this long...
It is too long, not too hard. Anyway I doubt anyone can do it in any exam without making a small mistakes ☺
it's like running a marathon (well, almost...)
Most elegant integral I've seen so far,thank you sir
Nothing even remotely elegant about this. This is the equivalent of opening a safe by dropping an anvil on it.
"When in doubt, use horseshoe mathematics."
You too saw the video!!
I see, you are a man of "NO U" culture here!
Easiest integral on RUclips
31 min yt video - "no i don't think i will"
do the integral of sqrt(tan(cuberoot(tanx)))
Kyle Farias Just use numerical integration on given bounds
Integrand has a period of pi. It's real-valued from [0,pi/2) and complex-valued from (pi/2,pi). Goes to +inf at pi/2.
Running a marathon is easy compared to solving this integral 🤓
That was a really enjoyable lil’ video. Thank you, it was a nice little refresher! 🙂
pure beauty...!
Thank you!!!!
Here is a simpler and reliable way to do any partial fraction, t = A (t^2-t+1) + (Bt+C) x (t+1) = (A + B) t^2 + (-A + B + C) t + (A + C). Therefore, A + B = 0 = A + C and -A + B + C = 1. Therefore, B = -A = C = 1/3. This is instead of plugging random values.
yea dog imma need you to check by differentiation
Very thorough explanation. As an idea for another integral video, this is one of the most difficult integrals I ever got in college :
Integral [ 2x^3 / ( ( x^6 + 3 ) ^ (1/2) ) + ( 3x^2 / (4 - x^2) )^(1/6) ] dx
From x=0 to x=1
This could be the "scariest integral on RUclips" ;)
It is like fx+f(inverese) x so ans will be 1*1
When title says easy and time says half an hour
🤔🤔🤔🤔🤔🤔🤔😱
Cringe
It's also expressible in terms of "the" hypergeometric function 2F1.
Dr. BalckPenRedPen, please check the answer by differentiation :)
An excellent video, explained clearly. Really enjoyed watching the video and even had a try myself after watching. Great job.
我非常喜欢这个视频关于积分。黑笔红笔老师的解释清楚！看到了这个视频我自己试一试解决这个积分！
....It wasn't that bad....it didn't have limits to substitute (hehe ! )
My friend had given me this integral as a challenge and I had solved it. I would say this a great algebraic integration problem.
now the nth root
Tumbs up for this ace demonstration !!!
Can you do integral of (xtanxdx) ?
It looks so innocent at the very beginning. lol
Small technical error:
In the final function, you used the constant C to represent the constant of integration. BUT, you already used C to represent a different constant in your partial fraction decomposition. So, the final solution should use a different constant, like C-sub-1 or K.
Wolfram gives a general solution in terms of hypergeometric functions:
int [tan(x)]^a dx = 2F1(1, (a+1)/2; (a+3)/2; -tan(x)^2) . ([tan(x)]^(a+1))/(a+1) + C
In effect, a hypergeometric "correction" factor applied against a power rule-like construct operating on tan(x), as though you naively integrated it. Hence
int [tan(x)]^(1/3) dx = 2F1(1, 2/3; 5/3; -tan(x)^2) . ([tan(x)]^(4/3))/(4/3) + C.
which to me is a lot nicer. One of the things about maths is that there isn't one "true" or "best" _representation_ of a mathematical object, but unfortunately, the way it's taught conveys this idea that there is.
...And when you want to take a break from listening to this video, you are going to SUBSTITUTE IT for one of the easier/more beautiful ones to understand :D Like sin(z)=2
For anyone wondering, the constant to make them equal is pi * sqrt(3) /12
Okay so you've done sqrt tan x, you've done cube root tan x, can you generalize? Nth root tan x ?
"and then we have to get back to the U world, and then we have to go back to the X world."
What a wild ride - we're going on interplanetary journeys solving this integral!
It kinda bothers me that you write "tan^-1" instead of "arctan", it makes it seem like you could get tan^-(1/3) by multiplying it with tan^(2/3).
Florian Bender if you mean 1/tan x, he would write it as cot x
Florian Bender tan^-1 is a standard notation for arctan
I agree. It's bad form IMO to both use tan^-1(x) for arctan(x) as well as the tan^q(x) for (tan(x))^q within a single exercise.
Monzur Rahman We're aware, but it's a terrible notational standard.
It's not even a "standard" notation really. Just a commonly used one (and not even as commonly used as the arctan notation).
Wolfram Alpha says" you goddamn right boy"
You can't impress a girl with it
sure it's too smart for them....
I disagree. Perhaps your trying to impress the wrong girls.
No you can't impress the girls quite true, but you can impress them with the money you make if you're working in this particular field.
If you can't impress your girl with this stuff...
*you're not talking to the right girls*
Almost all girls like money, luxury cars and spending time shopping. They want M-O-N-E-Y......and their math skills is +....+money+money+money......
You're videos are awesome! Please keep posting :)
Everytime he says 'something' I hear 'sensei' *_*
Samarth Varshney or... passdee instead of positive. Forgive him, neither him or me are english ppl
Not even a single mistake . Great job bro