@blackpenredpen now solve Integral {[(e power sinx + sqrt(cosx+tanx)] power pi} ---------------------------------------- [ln (ln (cosec x)) + sqrt(sec²x+cot²x)]
One professor I had used to tell us that: "in the test we usually put some "ghosts", my recommendation, don't get scare for those "ghosts" at the end they are really easy to deal with"
@@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.
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.
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 :)
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 😀
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!!
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!
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
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.
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".
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!
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)
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.
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?
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.
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.
@@MemorizadordeVerbos someone said in comments: "it's what you write when you don't have defined bounds for an integral. Given that the derivative of a constant is 0, that means there are an infinite amount of constants that could exist. So we write + C to say "including any constant that was 'lost' due to the derivative of a constant being 0""
@@MemorizadordeVerboscause if you differentiate it ,you can reach same integration value with any +c or -c ,so it has infinite possibilities,to show that we write +c as a convention (I don’t know wheather convention is right word but yes similar to that)
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 💀
One professor I had used to tell us that: "in the test we usually put some "ghosts", my recommendation, don't get scare for those "ghosts" at the end they are really easy to deal with"
i did not expect such a clean solution for this integral
It was way too easy question though
@@GokuDragon-d4rayyy 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 👍
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
“Everybody’s here”
I love that part.
smash bros but it’s an integral
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"
There is nothing more satisfying than terms that perfectly cancel/match out, leaving a neat answer!
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
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.
You are so calm, smart and good natured I can't help but smile as I watch you solve these. Good work!
When i saw this, it sort of look solvable and now i see this being solved makes it so satisfying
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.
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 person learned basic integration only yesterday, this thing really is fascinating
This integral looks so daunting to complete, and watching it quickly simplify down is just really satisfying
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
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 😀
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!
That was just beautiful. I've never seen such a complicated integral become so easy in such a low amount of time.
it feels so blessed to understand this after taking calc bc
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 😊
I'm just impressed he managed to fit it all on the whiteboard.
This is one of the most beautiful integrals I've ever seen...
I have not got the faintest idea what he just said, but the way he explained everything made me feel like I did
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
@@R.EDACED6year 8 is like 14 right? Too old for calc 1. Im my country we have math phd by age 16
Didn't expect such a easy solution
This was a fun one to do! Cool how all the identities and the u-sub lines up nicely
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!!!
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 is the most beautiful integral I’ve ever seen
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
When saw the video where you made the question I actually tried to solve it and it took solid 10 mins before I got the answer.
The fact that I had to play this video at 2x just gives away how easy that question actually was
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! 🙂
3:20 thats not true if sin(x) is negative, we were lucky to have ln(cosec(x)) in the denominator
5:46 I thought he is going to say "pi is just 3"
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".
NO way how did it fit so perfectly
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.
This is possibly the greatest integral I've ever seen!
I just know basic trigonometry and nothing about calculus but it was satisfying
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!
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
This video is great for revising calculus rather than just fun purposes.
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.
6:18 and where did the chain rule go for sec(x)
What are you talking about?
Why does -2cotx on the denominator become just -1/2 on the top? What happened to the cotx?
4:07
he rewrote the 1/cotx as tanx in order to bring it to the top :)
@@square.banana ohh I didn't notice that
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.
I love it how brother gets all gitty to do this problem and then starts cranking 90s with the dry-erase marker.
Best math teacher. You are incredible
Don’t forget the plus c!
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?
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.
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
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
This was a very interesting one to solve
Loved it! :D
🗣️ : "This joke is frankly getting kind of stale"
Me : "WHAT?!?!" *STONE COLD INTENSIFIES*
I thought you couldn't use the power rule, if the exponent in the log is less than 0. correct me if I'm wrong plz 3:21
It's astonishing that taking calculus can allow me to understand this even a little bit
love this one, thank you
i like how only sec x remains like the others were murdered lol
Ohh so just add random letters until it’s solved! Just amazing 😭😭😭
Pretty much 😆
I am pretty sure the product rule uv'+vu' 4:10
3:45 isnt the derivative of any constant just 0?
6:18 Not done the +c 😂😂😂
Now do one with all of the trig functions including the inverse and hyperbolic trig functions.
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.
Nicely solved!
I love how there are black and red whiteboard pen boxes in bulk under his table 🥰
that double tap of pen made me scared, but i mustered courage to face it
Can you explain the vibrating string string 2nd order PDE in a video
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.
I don't understand any of these but I still watch ur vids
This was a beautiful integral
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.
I have a question. You use ln(sin(x)^-2)=-2ln(sin(x)). I think this is only valid if sin(x)>=0 but not if sin(x)
These are the type of questions which should be left without a thought
wow its so much easier than i thought
It turned out really easy
beautifully explained
I got everything! nice job
I’m too late for this but ever considered adding sinh(x) cosh(x) tanh(x) coth(x) etc, to the integral?
Extra Credit: Make lower limit zero and solve for upper limit that makes the definite integral equal to 1.
This came out on my birthday and I didnt even notice it 😂
This looks so hard but is so satisfying
Hi, just found your channel, only 1:29 into the video and I can't get past the absolute whiteboard chad moves you're doing with the marker swapping
I like your weird words magic man
funny*
Why not add some hyperbolic trig while you at it
6:20 "PLUS C, YAY!"
maths is all about approach
everybody can learn fromulae and concepts but approach comes with practice and skills.
Try doing any integral to the e power.
6:20 Where did that +c came from?
It's just integral constant
@@Sanatanirohit._ IK but why?
@@MemorizadordeVerbos someone said in comments:
"it's what you write when you don't have defined bounds for an integral. Given that the derivative of a constant is 0, that means there are an infinite amount of constants that could exist. So we write + C to say "including any constant that was 'lost' due to the derivative of a constant being 0""
@@MemorizadordeVerboscause if you differentiate it ,you can reach same integration value with any +c or -c ,so it has infinite possibilities,to show that we write +c as a convention (I don’t know wheather convention is right word but yes similar to that)
I have a question regarding trig , why does the cos(x-π/2) = sin(x) but cos(x-90) doesn’t equal sin(x)?
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 don't know I of integration but it's always fun to watch people solving it 😅
as a math 100 student, can confirm this was on my midterm