@blackpenredpen now solve Integral {[(e power sinx + sqrt(cosx+tanx)] power pi} ---------------------------------------- [ln (ln (cosec x)) + sqrt(sec²x+cot²x)]
@@pitoachumi2663 bro u r so right. I just started calculus and it is already hard. But the truth is this one was extremeley satisfying. But the truth is that in India the most difficult questions would be very easily solved by the teachers but we end up gettting dumbfounded by the same questions. It's annoying but fun.
If you love physics solve The speed of a motor engine decreases from 900 rev/min. to 600rev / min in in 10 seconds. Calculate: The angular acceleration Number of revolutions made by the motor during this interval How many additional seconds are required for motor to come to rest in the same rate
If you love physics solve find the value of ratio of specific heat capacity for the mixture of gas containing one mole of nitrogen and 2 mole of argon if Gamma of nitrogen is equal to 1.40 and gamma of argon is equal to 1.54
If you love physics solve 3 moles of an ideal gas undergoes a reversible isothermal compression at 27 degree Celsius during this compression 1850 joule of work is done on the gas what is the change in entropy
Time to try it out! sin²+cos²+tan²=1+tan²=sec² (1-2ln(cos))^π ln(csc)+ln(sec)+ln(cot) = -ln(sin)-ln(cos)+ln(cos)-ln(sin) = -ln(sin) d/dx(-2ln(sin)) = -2 × cos/sin = -2cot So the integral becomes sin×(1-2ln(cos))^π / -2cos u = -ln(cos) du = sin/cos dx So (2u+1)^π du Where u = -ln(cos x) Now another substitution w = 2u+1 dw = 2du So -¼×w^π dw Integrates to w^π+1 / (-4)(π+1) w = 2u+1 u = -ln(cos x) w = 1-2ln(cos x) (1-2ln(cos x))^(π+1) / -4(π+1) + C
5:12 My favorite scene because we all know what he means 😂 This problem was actually quite easy for looking so difficult. The only things I didn't instantly see is the u-sub, because I didn't instantly know the derivative of sec(x) Great video!
I love how real the problem creation is. Start with a random smattering of operations and functions that look scary then adjust it until a solvable problem appears.
True, I love when they teach you the absolute basics and then immediately give you a classwork on the toughest parts without the chance to even study 😀
please make more "insane" looking integrals that are actually fairly easy! I loved the way you break down the solution for something so ridiculous looking :)
as a 50 year old I recently(2020-2022) went back and got my associates degrees after a debilitating health incident that left me reconstructing my base of knowledge. I also wanted to work my brain and ended up taking physics and calculus 1 and 2. At the community college level. I still have memory issues but these videos keep me atleast familiar with the concept of integration. I did well almost coming out with an A but for fractions of points in both.. graduated with a 3.92 gpa. the final Bs in calc I and II got my 4.0.. I can now help my highschool and middleschool children when they get stumped in their advanced math classes. I love the depth to which you dive in explaining your process and the sloutions..Your humor is also not lost on me. Fun makes learning more bearable in difficult situations. I only wish my professor was so concerned with teaching the process. After the few example in the book, she assigned out homework. She wanted it the way she taught it but left us to "figure it out" no pun intended.. I would have drowned but for math is power, professor leanard and some of the others. I realized the value of not only knowing but proving the trig identites alittle too late but now I know. I may go back and finish the bachelors in meterology but am leaning toward mathematics just for fun. Keep up the good work, you are opening doors in the minds of many who were traumatized by higher maths as evidenced by the comments.. my only question is whether all the variations of blackpenredpen is actually you? there are several, its hard to tell these days.
I’m a high school student rn and I’ve been watching blackpenredpen for a few years now. Well I’ve finally made it to calculus in school and this is the first time in my life I’ve actually understood what happened in the vid (normally id just watch and be interested), pretty cool!
BlackpenRedpen I really am your fan and i love your videos, your way of explanation fascinates me and I understood every topic of your video, even my mother encourages your teaching and I had watched almost all your videos and I still watch them! Thank you for your teaching and making math enjoyable I had learnt many things from you, and I encourage you to keep going! Al the best!!
As a grade 12 student myself, I didn't find this too hard. Kinda proud of myself that I am able to do such calculus probelms, with mich ease. Loved the vid.
@@RJiiFin Say whatever you want man, I don't care. I solved the question and I'm proud of it, and no, I don't find it weird to flex about it as a grade 12 student.
It's like one of those jokes where you can tell he started with the punchline and worked his way back... Problem for students is that if you're stressed out and forget just one of those trig relationships, you're screwed. 😛
You only need one, sin2+cos2=1 and it can be derived from geometry on the fly. Did that more than once in high school. Trig functions are much easier than some integration tricks and substitutions (the hardest part in this solution frankly)
The function becomes undefined every (npi/2, n2pi), and the limit of that function as it goes to kpi+(2pi - pi/2) (where k is an element of odd natural numbers) goes to negative infinity and finding the area from a to b of fx under the curve and a , b are discontinous (like they belong to different sections of graph and are being separated by a discontinuity) will always be -inf so to get a determinate area a,b has to be continous
This is the kind of stuff that you throw into Wolfram Alpha when the teachers no longer care about the exact solution steps for integrals. That was a really pleasant discovery for me when I started the math courses after calculus, when the teachers started being like "just find the numerical solution in some way".
I got scared for a second and thought it was 1 + (ln(sin^2(x) + cos^2(x) + tan^2(x)))^pi but then i realized it was the whole top raised to pi and a couple trig identities later the answer presented itself. Love it.
We completed indefinite integration in our highschool yesterday and first thing in my mind was to visit bprp... And I did solve it on my own before watching his solution. Yay!
They're actually super simple! If you're trying to find the derivative of x^2+y^3+z^4 Partial derivative of x -- simply treat Y and Z as constants, so you get 2x. That's it!
It's not easy. They're very abstract, so methods work for them even less than for regular ODEs. I guess one trick is assuming separability, and seeing what transpires. So you have a function f(x,y,z). Maybe you're solving something like f_xx=f_yz. Something that looks separable. Just assume f(x,y,z)=X(x)Y(y)Z(z) and see if you can derive a set of solutions.
This problem rules, and I love that I'm far enough into calculus to be able to not only follow along, but (possibly) solve this type of integral. The only 2 parts I was worried about were taking the derivative of the denominator while leaving it as 2ln|csc|, but it was simple to still get -2cotx. The other part that I'm not sure if I could find is what to set as u, my professor has been very generous with what u is going to equal, but looking back at this problem I would likely originally try setting just secx as u, finding that doesn't work, and then hopefully try setting everything within the parentheses, but we'll see. What strategy would you recommend when trying to find u? I have my exam over this section Tuesday, wish me luck!
Looking back at this with some actual sleep... All this is is a u-sub (and a lot of simplifying) lol. All the trig threw me off since we're doing trig sub and fraction decomp. right now. It's insane that cal 1 could do this, I'm definitely sharing this with my previous professor
also him not putting absolute values when doing ln(csc^2(x)) = 2ln|csc(x)| = 2ln(csc(x)) is because the integrand restricts csc(x) > 0 by nature similarly with ln(sec^2(x))
Coming from a high school student who just learned integration, my teacher recommended I pick an inside function as u, and if the derivative is seen in the expression in the integral, that most likely is u. Again, I might be completely wrong.
That was so freaking cool. The best part is that even though it's so complicated, a high schooler should be able to do that if they have taken precalculus and calculus!
I always feel uncomfortable when even power (2 in this case) is factored in front of ln, while since then the contents of the ln may turn (and we know for sure that it does so in case of trigonometric functions), so strictly speaking you have to put a modulus (abs. value), so that ln (y^2) = 2 ln |y|. But I constatntly see this thing like many alike are neglected while taking integrals, I wonder does it always works? If yes, why, If no, when it works?
It's interesting how some integrals on paper look very complicated but you only need calculus 1/AP Calculus AB skills to solve the entire integral, albeit with a lot of steps.
There is a reason the hardest part of calculus is algebra. There is nothing intuitive about this. It's like learning the grammar of a language. You just have to memorize enough rules to be able to solve the equation. The only way to actually understand the relationships being invoked is to also be able to prove each and every rule. This is why studying for the subject is such a brute force labor intensive process.
If you don't understand what a sine or cosine are, then yes, this is just memorization. If you are patient with yourself and remember what a sine, cosine, etc represent, then you can go back to first principles and figure out what it is you need to do. You get past rote memorization by understanding the fundamentals, so when you see cot^2x you don't lose your mind. It's just another ratio, it has substitution principles, etc.
Disagree. While calculus requires you to memorize the foundations of algebra, it still requires intuition for you to manipulate things to make them usable. Just like a puzzle, just because you have all the pieces doesn't mean you can put it all together without intuition.
@@mainsera4407 intuition is bullshit, just gonna come right at that garbage. inuiting something means you don't know why you know, you just do. That has no place in mathematics at this level.
I don't know about the "nothing intuitive" part... It reminds me of learning programming languages, at first it's just reading examples + memorization, but when you got a couple such languages on hand you start to recognize certain patterns & ease of supposedly remembering the syntax and so on...
Where is the hyperbolic and the inverse of all the functions (sin, cos, tan, csc, sec, cot, arcsin, arccos, arctan, arccsc, arcsec, arccot, sinh, cosh, tanh, csch, sech, coth, arcsinh, arccosh, arctanh, arccsch, arcsech, and arccoth). Then you will have every trigonometric function in one integral. Here is one: Take the indefinite integral (Using the complex definitions of the trigonometric functions to get values for all x) of: sin(x)^(tan(x)) - cos(x)+csc(x)sec(x) - arcsin(cot(x))/(arccos(arctan(x)) -sqrt(sech/arccoth) + (arccsc(ln(x)))/(sinh(cosh(arccot(tanh(x)-arcsec(x)))))+csch(π*x^(arcsech(x)-coth(x)) + arcsinh(x) - W(arccosh(arctanh(sqrt(x)))) + arccsch(x) + arcsech(x) + arccoth(x) 🤣
@@tombratcher6938That's just how u-sub works. You pick a portion of the integral that differentiates into another portion of the integral. If it doesn't work out cleanly, you do differentiation by parts instead.
I dont understand which trig functions are which here because i do mathematics in greek but the weirdest part is that i understood this somewhat and it baffles me
Learn more calculus on Brilliant: 👉brilliant.org/blackpenredpen/ (now with a 30-day free trial plus 20% off with this link!)
Thank you for this information but I really want you to tell some basic for integration thanks you
Thanks from Brazil!!!
🙏♾🎁👏🏻👏🏻👏🏻🎉🎊
@blackpenredpen now solve
Integral
{[(e power sinx + sqrt(cosx+tanx)] power pi}
----------------------------------------
[ln (ln (cosec x)) + sqrt(sec²x+cot²x)]
Nice what is your country name
India
What the student really needs to learn here is not Calculus, but to control their own panic.
Exactly! It is natural to feel intimidated and want to reach for the biggest weapon, but this beast can be taken down barehanded.
Biggest thing I realized in calculus is it's not that bad if you master trigonometry
Exactly! Like at home sure i have all the time but at the exam I'd feel the cold sweat dripping down my face fr 💀
i did not expect such a clean solution for this integral
It was way too easy question though
@@user-qz6sh7dy2hayyy same pfp
@@canyoupoopwot da fok
@@canyoupoop YO LUFFY wyd learning calc bro😭😭😭
thats the beauty of math!
As a high school cal student, this is, so far, the most satisfying integral I've ever witnessed.
really?
come to india baby ull find satisfying triggers everyday then
@@pitoachumi2663 oooo
@@pitoachumi2663or just use the internet
You better work in your tech support @@pitoachumi2663
@@pitoachumi2663 bro u r so right. I just started calculus and it is already hard. But the truth is this one was extremeley satisfying. But the truth is that in India the most difficult questions would be very easily solved by the teachers but we end up gettting dumbfounded by the same questions. It's annoying but fun.
It’s always so funny to me when you add the sin(x)^2 + cos(x)^2 just to make it look harder but it’s just 1 😂 I love it lol
If you love physics solve
The speed of a motor engine decreases from 900 rev/min. to 600rev / min in in 10 seconds. Calculate:
The angular acceleration
Number of revolutions made by the motor during this interval
How many additional seconds are required for motor to come to rest in the same rate
If you love physics solve
find the value of ratio of specific heat capacity for the mixture of gas containing one mole of nitrogen and 2 mole of argon if Gamma of nitrogen is equal to 1.40 and gamma of argon is equal to 1.54
If you love physics solve
3 moles of an ideal gas undergoes a reversible isothermal compression at 27 degree Celsius during this compression 1850 joule of work is done on the gas what is the change in entropy
@@DiverseDose11i was literally revising this exact topic ill solve it when i finish revising
@@NiceLol-dl6lq yeah it will be nice 👍
“Everybody’s here”
I love that part.
smash bros but it’s an integral
What a glorious result! I'm going to make this extra credit the first time I teach a calculus class!
😆
Epik
All your students who watch blackpenredpen: STONKS!
@@caroot1085 If they watch this, they deserve the score
Time to try it out!
sin²+cos²+tan²=1+tan²=sec²
(1-2ln(cos))^π
ln(csc)+ln(sec)+ln(cot)
= -ln(sin)-ln(cos)+ln(cos)-ln(sin)
= -ln(sin)
d/dx(-2ln(sin)) = -2 × cos/sin
= -2cot
So the integral becomes
sin×(1-2ln(cos))^π / -2cos
u = -ln(cos)
du = sin/cos dx
So
(2u+1)^π du
Where u = -ln(cos x)
Now another substitution
w = 2u+1
dw = 2du
So -¼×w^π dw
Integrates to
w^π+1 / (-4)(π+1)
w = 2u+1
u = -ln(cos x)
w = 1-2ln(cos x)
(1-2ln(cos x))^(π+1) / -4(π+1) + C
Lmao it has the translate to english button
@@R8Spike and I got Hindi button &
sine = sin (pronounced seen) lol
why didn't you take the w as 1-2ln(cosx) from the beginning it's easier than two substitutions
@@mandarbamane4268 for me it was sine = syn
@@TRT_MOOSIC for me it translated sin to "paaap"
This is the only guy that can make math fun. Keep up the amazing content!
Thanks!!
5:12 My favorite scene because we all know what he means 😂
This problem was actually quite easy for looking so difficult. The only things I didn't instantly see is the u-sub, because I didn't instantly know the derivative of sec(x)
Great video!
Stuff like that is why I keep watching this guy
idk what he means
What does he mean
There is nothing more satisfying than terms that perfectly cancel/match out, leaving a neat answer!
I love how real the problem creation is. Start with a random smattering of operations and functions that look scary then adjust it until a solvable problem appears.
When i saw this, it sort of look solvable and now i see this being solved makes it so satisfying
It’s always good when teacher give you things in a test you’ve never seen before, like cscx, secx and cotx
True, I love when they teach you the absolute basics and then immediately give you a classwork on the toughest parts without the chance to even study 😀
You are so calm, smart and good natured I can't help but smile as I watch you solve these. Good work!
please make more "insane" looking integrals that are actually fairly easy! I loved the way you break down the solution for something so ridiculous looking :)
as a 50 year old I recently(2020-2022) went back and got my associates degrees after a debilitating health incident that left me reconstructing my base of knowledge. I also wanted to work my brain and ended up taking physics and calculus 1 and 2. At the community college level. I still have memory issues but these videos keep me atleast familiar with the concept of integration. I did well almost coming out with an A but for fractions of points in both.. graduated with a 3.92 gpa. the final Bs in calc I and II got my 4.0.. I can now help my highschool and middleschool children when they get stumped in their advanced math classes.
I love the depth to which you dive in explaining your process and the sloutions..Your humor is also not lost on me. Fun makes learning more bearable in difficult situations. I only wish my professor was so concerned with teaching the process. After the few example in the book, she assigned out homework. She wanted it the way she taught it but left us to "figure it out" no pun intended.. I would have drowned but for math is power, professor leanard and some of the others. I realized the value of not only knowing but proving the trig identites alittle too late but now I know. I may go back and finish the bachelors in meterology but am leaning toward mathematics just for fun. Keep up the good work, you are opening doors in the minds of many who were traumatized by higher maths as evidenced by the comments.. my only question is whether all the variations of blackpenredpen is actually you? there are several, its hard to tell these days.
I'm in high school and this year we finally did the indefinite integrals. Now I can comprehend some of this man’s videos. Nice solution
As a student that has no idea what half of those term means, it was enjoyable. I’ll take a look at this again once I learned those terms.
Your alway best. It seems to be very hard but after your explanation it looks like very easy one. Keep rocking on mathematicians
Thanks a lot 😊
it feels so blessed to understand this after taking calc bc
I have not got the faintest idea what he just said, but the way he explained everything made me feel like I did
what am i doing here? im a biology major
Understanding the beauty of integration
I’m a high school student rn and I’ve been watching blackpenredpen for a few years now. Well I’ve finally made it to calculus in school and this is the first time in my life I’ve actually understood what happened in the vid (normally id just watch and be interested), pretty cool!
The fact that I had to play this video at 2x just gives away how easy that question actually was
I'm just impressed he managed to fit it all on the whiteboard.
This integral looks so daunting to complete, and watching it quickly simplify down is just really satisfying
That was just beautiful. I've never seen such a complicated integral become so easy in such a low amount of time.
As a 8th grader I understand absolutely nothing but I like the way he is teaching
do you not do calc in year 8 in america?
@@user-oq7cx2rb4t I am not in America, in our country we learn calculus in grade 12 and college
Bro where do you live that you are doing Calc in year 8?
Instantly assuming he's from the us@@user-oq7cx2rb4t
@@_.Max.i.mus._year 8 is like 14 right? Too old for calc 1. Im my country we have math phd by age 16
BlackpenRedpen I really am your fan and i love your videos, your way of explanation fascinates me and I understood every topic of your video, even my mother encourages your teaching and I had watched almost all your videos and I still watch them! Thank you for your teaching and making math enjoyable I had learnt many things from you, and I encourage you to keep going! Al the best!!
Awesome, thank you!!!
This is one of the most beautiful integrals I've ever seen...
As a grade 12 student myself, I didn't find this too hard. Kinda proud of myself that I am able to do such calculus probelms, with mich ease. Loved the vid.
Well as a grade 11 student myself, I also didn't find this too hard. Kinda weird to flex about that as a grade 12 student though?
@@RJiiFin Say whatever you want man, I don't care. I solved the question and I'm proud of it, and no, I don't find it weird to flex about it as a grade 12 student.
@@RedditChronicles022You obviously do care because you responded 😁
@@RJiiFin Ok well, that's fair. Have a good day tho.
@@RedditChronicles022Thanks, you too! 🙂
It's like one of those jokes where you can tell he started with the punchline and worked his way back...
Problem for students is that if you're stressed out and forget just one of those trig relationships, you're screwed. 😛
You only need one, sin2+cos2=1 and it can be derived from geometry on the fly. Did that more than once in high school. Trig functions are much easier than some integration tricks and substitutions (the hardest part in this solution frankly)
I guess that's a problem for the student and a reason to git gud foe the teacher...
@@avelkm doesnt require geometry, just devide by cos^2 or sin^2
@@avelkm to be fair you won't have the time to start deriving some relationships on the fly during a test.
He’s high on pot-nuse
The function becomes undefined every (npi/2, n2pi), and the limit of that function as it goes to kpi+(2pi - pi/2) (where k is an element of odd natural numbers) goes to negative infinity and finding the area from a to b of fx under the curve and a , b are discontinous (like they belong to different sections of graph and are being separated by a discontinuity) will always be -inf so to get a determinate area a,b has to be continous
Didn't expect such a easy solution
I just know basic trigonometry and nothing about calculus but it was satisfying
I love how there are black and red whiteboard pen boxes in bulk under his table 🥰
suprisingly ..it was quite easy for me..i hit it right in the first time...nd....glad to see the answer matched ..as a beginner of integration.
This only works if you are in the first quadrant. Otherwise, you need to be careful about pulling the 2’s out of natural log.
but in the original integrand, we have in the denominator ln(csc(x)) ln(sec(x)) etc
which means it's restricted to first quadrant by nature
Best math teacher. You are incredible
This is the kind of stuff that you throw into Wolfram Alpha when the teachers no longer care about the exact solution steps for integrals.
That was a really pleasant discovery for me when I started the math courses after calculus, when the teachers started being like "just find the numerical solution in some way".
Watching this a month later, where I learnt a bit of diff and int calc, feels so nice to me bc I now understand what these are.
I got scared for a second and thought it was 1 + (ln(sin^2(x) + cos^2(x) + tan^2(x)))^pi but then i realized it was the whole top raised to pi and a couple trig identities later the answer presented itself.
Love it.
We completed indefinite integration in our highschool yesterday and first thing in my mind was to visit bprp... And I did solve it on my own before watching his solution. Yay!
This was a fun one to do! Cool how all the identities and the u-sub lines up nicely
Actually, not everybody is here... You forgot hacoversine and covercosine, for example. Those ones always get left out for some reason.
You could also include the arctrigonometric functions, the hyperbolic and arhyperbolic functions, even arg or atan2 could come into play.
Can you do content on partial differential equations (PDEs)
They're actually super simple! If you're trying to find the derivative of x^2+y^3+z^4
Partial derivative of x -- simply treat Y and Z as constants, so you get 2x. That's it!
@@mattreichmann8118 that's not what a pde is
It's not easy. They're very abstract, so methods work for them even less than for regular ODEs.
I guess one trick is assuming separability, and seeing what transpires.
So you have a function f(x,y,z). Maybe you're solving something like f_xx=f_yz. Something that looks separable. Just assume f(x,y,z)=X(x)Y(y)Z(z) and see if you can derive a set of solutions.
This video is great for revising calculus rather than just fun purposes.
Great video!
The trick here was to keep the sin in the denominator and the sec in the numerator by playing with the negative exponent
maths is all about approach
everybody can learn fromulae and concepts but approach comes with practice and skills.
Extra Credit: Make lower limit zero and solve for upper limit that makes the definite integral equal to 1.
This is possibly the greatest integral I've ever seen!
I love it how brother gets all gitty to do this problem and then starts cranking 90s with the dry-erase marker.
"a_generic_nerd" needs to be asked "STATE'S RIGHTS TO DO WHAT?"
You're telling me a guy with a confederate flag profile pic couldn't do math? I'm shocked, nay, flabbergasted by this information.
5:46 I thought he is going to say "pi is just 3"
NO way how did it fit so perfectly
Don’t forget the plus c!
Hyperbolic trig next?
Ohh so just add random letters until it’s solved! Just amazing 😭😭😭
Pretty much 😆
It turned out really easy
This came out on my birthday and I didnt even notice it 😂
my ass not understanding what the fuck he just wrote at the start: 👍
I don't know I of integration but it's always fun to watch people solving it 😅
I don't understand any of these but I still watch ur vids
I love these videos but I have absolutely no idea whether there's a system behind the pen colours or if it's just a semi-random choice thing.
It’s systematic. The red is for important parts, new parts or side notes.
🗣️ : "This joke is frankly getting kind of stale"
Me : "WHAT?!?!" *STONE COLD INTENSIFIES*
This problem rules, and I love that I'm far enough into calculus to be able to not only follow along, but (possibly) solve this type of integral. The only 2 parts I was worried about were taking the derivative of the denominator while leaving it as 2ln|csc|, but it was simple to still get -2cotx. The other part that I'm not sure if I could find is what to set as u, my professor has been very generous with what u is going to equal, but looking back at this problem I would likely originally try setting just secx as u, finding that doesn't work, and then hopefully try setting everything within the parentheses, but we'll see. What strategy would you recommend when trying to find u? I have my exam over this section Tuesday, wish me luck!
Looking back at this with some actual sleep... All this is is a u-sub (and a lot of simplifying) lol. All the trig threw me off since we're doing trig sub and fraction decomp. right now. It's insane that cal 1 could do this, I'm definitely sharing this with my previous professor
also him not putting absolute values when doing ln(csc^2(x)) = 2ln|csc(x)| = 2ln(csc(x)) is because the integrand restricts csc(x) > 0 by nature
similarly with ln(sec^2(x))
Coming from a high school student who just learned integration, my teacher recommended I pick an inside function as u, and if the derivative is seen in the expression in the integral, that most likely is u. Again, I might be completely wrong.
@@eshwarthammineni7911 this generally is a good tactic to try for some integrals.
Its real funny that the confederate flag pfp is getting pissy about it
Now I know what kind of extra task gonna be solved by my students during their exam tests😁😁😁
I am good teacher, I am good😝
I love how the integral symbol goes slowly losing its power and shrinking more and more lol
Why not add some hyperbolic trig while you at it
that double tap of pen made me scared, but i mustered courage to face it
Sir I have done this one by myself only and then I checked that mine was correct
Very happy to know that😊
That was so freaking cool. The best part is that even though it's so complicated, a high schooler should be able to do that if they have taken precalculus and calculus!
*With enough patience
This was a very interesting one to solve
Loved it! :D
I always feel uncomfortable when even power (2 in this case) is factored in front of ln, while since then the contents of the ln may turn (and we know for sure that it does so in case of trigonometric functions), so strictly speaking you have to put a modulus (abs. value), so that ln (y^2) = 2 ln |y|. But I constatntly see this thing like many alike are neglected while taking integrals, I wonder does it always works? If yes, why, If no, when it works?
Nicely solved!
It's interesting how some integrals on paper look very complicated but you only need calculus 1/AP Calculus AB skills to solve the entire integral, albeit with a lot of steps.
There is a reason the hardest part of calculus is algebra. There is nothing intuitive about this. It's like learning the grammar of a language. You just have to memorize enough rules to be able to solve the equation. The only way to actually understand the relationships being invoked is to also be able to prove each and every rule. This is why studying for the subject is such a brute force labor intensive process.
If you don't understand what a sine or cosine are, then yes, this is just memorization. If you are patient with yourself and remember what a sine, cosine, etc represent, then you can go back to first principles and figure out what it is you need to do. You get past rote memorization by understanding the fundamentals, so when you see cot^2x you don't lose your mind. It's just another ratio, it has substitution principles, etc.
Disagree. While calculus requires you to memorize the foundations of algebra, it still requires intuition for you to manipulate things to make them usable. Just like a puzzle, just because you have all the pieces doesn't mean you can put it all together without intuition.
@@mainsera4407 intuition is bullshit, just gonna come right at that garbage. inuiting something means you don't know why you know, you just do. That has no place in mathematics at this level.
I don't know about the "nothing intuitive" part...
It reminds me of learning programming languages, at first it's just reading examples + memorization, but when you got a couple such languages on hand you start to recognize certain patterns & ease of supposedly remembering the syntax and so on...
It's astonishing that taking calculus can allow me to understand this even a little bit
I would have forgotten the "+ C", as I always do.
beautifully explained
Anyone here this time
6:18 Not done the +c 😂😂😂
love this one, thank you
"Can we solve this? - Yes, because I made this!" 😂
These are the type of questions which should be left without a thought
I got everything! nice job
cscx = 1/Sinx
Now add in the hyperbolic and inverse trig functions 🙃
This was such a satisfying one to do on my own. The way everything cancelled as I used a couple trig identities and a u sub was so satisfying
Try doing any integral to the e power.
Where is the hyperbolic and the inverse of all the functions (sin, cos, tan, csc, sec, cot, arcsin, arccos, arctan, arccsc, arcsec, arccot, sinh, cosh, tanh, csch, sech, coth, arcsinh, arccosh, arctanh, arccsch, arcsech, and arccoth). Then you will have every trigonometric function in one integral. Here is one:
Take the indefinite integral (Using the complex definitions of the trigonometric functions to get values for all x) of:
sin(x)^(tan(x)) - cos(x)+csc(x)sec(x) - arcsin(cot(x))/(arccos(arctan(x)) -sqrt(sech/arccoth) + (arccsc(ln(x)))/(sinh(cosh(arccot(tanh(x)-arcsec(x)))))+csch(π*x^(arcsech(x)-coth(x)) + arcsinh(x) - W(arccosh(arctanh(sqrt(x)))) + arccsch(x) + arcsech(x) + arccoth(x)
🤣
i like how only sec x remains like the others were murdered lol
This was a beautiful integral
I’m too late for this but ever considered adding sinh(x) cosh(x) tanh(x) coth(x) etc, to the integral?
Can you explain the vibrating string string 2nd order PDE in a video
"This looks terrible, but just use this substitution which you pull out of nowhere and somehow happens to work"
What substitution was pulled out of nowhere?
That summarises my experience with maths lmao
@@JustBackgroundNoiseu = 1+2 ln(sec(x))
@@tombratcher6938That's just how u-sub works. You pick a portion of the integral that differentiates into another portion of the integral. If it doesn't work out cleanly, you do differentiation by parts instead.
@@tombratcher6938 when it comes to trig functions, u-sub is very likely to be in play due to the nature of trig functions.
I dont understand which trig functions are which here because i do mathematics in greek but the weirdest part is that i understood this somewhat and it baffles me
That was clean, i did everything fine except the u sub