tan(x - arctan(x)) + C 1. divide top and bottom by x² so we get 1/(sinx + 1/x * cosx)^2 2. factor out sqrt(1/x^2 + 1) inside the square to get (x^2/(1 + x^2)) * (1/(sinx*1/ sqrt(1/x^2 + 1) + cosx*(1/x)/sqrt(1/x^2 + 1))^2 3. use the formula for cos(a + b) to get (x^2/(1 + x^2)) * sec^2(x - arctan(x)) 4. let u = x - arctan(x) so that x^2/(1 + x^2) dx = du, so that the integral becomes integral sec^2 u du 5. Integrate to get tan(x - arctan(x))
@FolyPlays Expand cos(x-arctan(x)) using cos(a-b)=cosacosb+sinasinb, use cos(arctan(x))=1/sqrt(1+x^2)=1/x*1/(sqrt(1+1/x^2)) and sin(arctan(x))=x/sqrt(1+x^2)=1/sqrt(1/x^2+1)
Love these! Your explanations and way of explaining always help me to see clearly the logic and process. So much so that I was able to write the integral down and do it myself from scratch... all the way thru without looking for a hint. And I finished it in around 11 mins. awesome ! Thanks! I find these integrals strangely soothing... :)
u can also do this ques with substitution : x = tan t... in denominator u can get the formula of cos ( A-B) and with another sub after it we can get the answer
I actually solved it by dividing the numerator & denominator by (cosx)^2. Btw, this question was asked in our weekend exam. I'm taking the IIT-JEE exam on May-19-2019.
I've found out a more inquisitive way to solve it. use harmonic addition theorem on the denominator,and take the argument of the sine function(or cosine,depending on which one you prefer to use) as u,and differentiate.Also you'll have to to an easy partial fraction which immediately follows. Just try it!
Doesn't necessarily have to have constant coefficients. try to write the denominator as sin(x+arctan(1/x)) then take the argument of this sine function as u and see what happens
Since the order of the denominator of ○'/○^n decreases and becomes simpler when integrated, it may work well to use it as the integrating side of a partial integration.
Recipe for crazy integrals: 1. Generate some crazy function 2 differentiate it. 3. Slap an i integral around the derivative . 4. Unleash on students. Watch the fun.
Love the integral and enjoy your way of teaching ! Always a pleasure to watch and listen ! But this intergral should not be in your calculus 2 EXAM but in a homework that could be evaluated . :D
X sqared means a regular square pattern. Xsinx means multiple waves. cosX added means inverted angles or I. Squaring multiple waves and Angeles ninety means absolute values of waves instances. When you divide one by the other and integrate you get circular ripples.
Yes but he did not finish Euler's substiutions which are closely related to Weierstrass substitution He did not finish Euler substitution because he showed one but there are three of them He has also Ostrogradski method for isolation rational part of integral which may be useful after Weierstrass or Euler substitution
Thanks sir I'm an JEE aspirant. My exam is on 2019.❤Thanks for the video, I'm studying at 3:00 AM this came in my recommendation and helped a lot. Btw which country are you from?
I’m only 16, I think we raise every factor inside the parentheses which will then be the indefinite integral of x squared over x squared times the square of sine x plus the square of cosine x. Sin^2(x) + cos^2(x) = 1, so the equation becomes: x^2 / x^2 * 1, any number multiplied by one stays the same. The equation becomes the indefinite integral of x squared over x squared and then the indefinite integral of 1 dx which results in x plus the integration constant. x + C We don’t learn about calculus yet, so this could be horribly wrong and stupid.
I think we can solve it by substitution also. 1. Divide by (cosx)^2 on numerator and denominator. 2. In numerator, yyou get(xsecx)^2 and in denominator you get (tanx +1/x)^2. 3. Now we can see that derivative of tanx is present in numerator. Well I might be wrong but can anyone please help me solve this further than this.
My answer is tan ( x-arctan(x)) + c I hot this by converting the denominator to one cos And this is the same answer if you use the formula for tan the diffrence between two angles and then convert tan to sin/cos
Another way from 2021 :) Bring the numerator to 1 via dividing by x^2, and compress the expression in the denominator via trigonometry. You should get ( x^2 * cosec^2( tan-1(1/x)+x ) ) / (x^2 + 1). Then you can directly substitute for tan-1(1/x)+x = u, and get the solution. Btw, when I saw this the first time, I didn't see that the denominator had a whole square lol. Lemme know if I missed something, though I hope I haven't.
I felt this was more of a guess and differentiate question rather than integrate. d(u/v)/dx= (v*du-u*dv)/u^2. Now if you'd notice the denominator is in terms of square. Rest of it is just finding v and v' which can be guessed with not too much efforts. If you want to call it a scam of a solution, its alright but this gets stuff done fast which is required. Disclaimer: You get to develop the intuition for the method only by practicing hard. So current JEE students should practice more as compared to cheats like this.
Isn't it also possible to simply do a complex partial fraction decomposition? It may not get easier, but the bruteforce method may save time in exams, if one can't find the trick 🤔
This is so famous and becomes a cakewalk with two substitutions that my teacher pointed out: x=tan θ and then (tan θ - θ) = t you'll get the answer straightway
We are taught this by putting the denominator = t and differentiating. And replacing values its more easy that way but i like the way you make it complex and detailed😃
Can u try the function : Y=sqr(cos[x])*cos(∞x)+sqr(x*0.00000001) and Y=sqr(cos[x])*cos(∞x)+sqr(-x*0.00000001) This couled be function with I. Together its like heart
And the big coincidence is that this video is uploaded almost 11 months ago and I usually like to watch PUBG mobile gameplay videos and luckly this video opened in my UA-cam homepage.
@@siddharthbhatt7752 i was on class IX dude.I mean it's not a big deal at all but it was one of the hardest problems i solved during the inception of my integral course
Have a stupid question perhaps. In order for the under the integral part to be equal when you multiply and divide by cos(x), don’t you need to specify that cos(x) is not equal zero? Otherwise the equation is false.
Well that was pretty easy If we take g(x) = xsinx + cosx Then the function under integration brings to mind the derivative of the quotient. It's quite easy to figure it out that if we take f(x) = sinx - xcosx then f'(x) = xsinx and f'g - fg' = x^2 and thus integral is just f/g + C And no, I'm not from India, I'm from Poland ;)
Zdaje się że ta całka pojawiła się w jednym z komentarzy pod innym filmem Stevena (wcześniejszym niż ten) i ja przedstawiłem wtedy sposób liczenia identyczny jak na tym filmiku Teraz już tego komentarza nie mogę znaleźć ale pamiętam że miałem taki sam pomysł na tę całkę co pokazany na filmie
I remembered this integral appearing in one of the integration competitions in China. The solution just says the answer is the answer because its derivative is equal to the given function.
Hey. .. just do a subsitution x =tanc. And it will be solved without by parts. ... I am preparing for jee .exam .. it's a tricky one . But by this subsitution it's it is easily solved. In denominator it becomes. sin sin + cos cos. And then we are done.....😊
There is a positive and a negative effect if you put this in your calc2 exam the positive is that your viewer count will increase because your students will watch this the negative is that your students will just memorise the answer
Right I'm a class 12 CBSE student. I have done this problem preparing for boards. I'm what people would call a "good" student( above 90%). I hated CBSE and it's method of testing. Honestly, I don't feel like there is a point in rushing students in an exam to solve complicated problems. How does it matter whether you solved it in 2 minutes or 5 minutes? The thinking process is really important.
Once again a great video by blackpenredpen 💘💖 but i have to say that Its not at all hard the same integral was asked in CBSE class 12 compartment maths board exam 2012 . So its a board level integral not even JEE main level. but yes it definitely required some creativity. (board exam means the final exams given by high school students in India. equivalent to advanced placement courses in us or Cambridge a levels, as levels in uk)
wow he solves it in 12 minutes but gives us only 2 seconds to try it.
hahaha : )))
You're supposed to pause the video and try it on your own ~
@@sawmill6358 It's a joke bro
He has to explain it too...😐😐
pause!
He never forgets + c .
Quite impressive .
i ALWAYS forget it
My auto correct always adds it
😂😂😂
EXTREMELY impressive
@@dqrksun yo.
tan(x - arctan(x)) + C
1. divide top and bottom by x² so we get 1/(sinx + 1/x * cosx)^2
2. factor out sqrt(1/x^2 + 1) inside the square to get (x^2/(1 + x^2)) * (1/(sinx*1/ sqrt(1/x^2 + 1) + cosx*(1/x)/sqrt(1/x^2 + 1))^2
3. use the formula for cos(a + b) to get (x^2/(1 + x^2)) * sec^2(x - arctan(x))
4. let u = x - arctan(x) so that x^2/(1 + x^2) dx = du, so that the integral becomes integral sec^2 u du
5. Integrate to get tan(x - arctan(x))
WOW!!!!!
@@blackpenredpen sorry I am late but can you explain 3rd line please?
@FolyPlays Expand cos(x-arctan(x)) using cos(a-b)=cosacosb+sinasinb, use cos(arctan(x))=1/sqrt(1+x^2)=1/x*1/(sqrt(1+1/x^2)) and sin(arctan(x))=x/sqrt(1+x^2)=1/sqrt(1/x^2+1)
@@folyplays-getgamified3613 cos(a+b)=cosa cosb-sina sinb
@@appybane8481 he knows that ofc he's talking about the line not the formula
Love these! Your explanations and way of explaining always help me to see clearly the logic and process. So much so that I was able to write the integral down and do it myself from scratch... all the way thru without looking for a hint. And I finished it in around 11 mins. awesome ! Thanks! I find these integrals strangely soothing... :)
Thank you!! : ))))
Lol, one of my friend ask me the Chinese WiFi problem and I answer immediately pi lolol
LOL
Lol
How many digits though?
Lol
@@ShadicgunMan 6-7 maybe doesn't matter you can solve using 355/113
u can also do this ques with substitution : x = tan t...
in denominator u can get the formula of cos ( A-B) and with another sub after it we can get the answer
"Let's do some math for fun"
Ummm, don't we always do math for fun?
Math On The Go totally!!!
ua-cam.com/video/y_XwQkchwrE/v-deo.html
6:19
bprp: so, we're gonna stop right here...
youtube: *An error occured, please try again later*
I say those are magic pens that know how to do calculus. I want those pens.
Honmestly impressed by the clear explanation and the fact that you went over all the passages.
Also really great attitude
I hope my sister will not find your video. 😂😂😂 I will give this as her exercise 😏😏😏.
marlon brade lol, nice!!
It’s only impossible when more than two pens are required
ua-cam.com/video/y_XwQkchwrE/v-deo.html
LIATE (logarithm-inverse trig-algebraic-trig-exponential): a powerful tool but not always the best
We were taught ILATE instead of LIATE.
Isn't it ILATE and not LIATE
Maybe, if you don't mind sounding like the White Rabbit from Alice in Wonderland (-; one pill makes you larger.... ;-)
It's ILATE
we are taught ilate .
Integration by parts for quotient. Damn. Clever.
Yay!!
It doesn't sound good. May be you get the answer .
We did this problem in the 12th class, so...
Of course,I'm from India.
we got the india part from your name.
@@julu2731 so funny hahahahaha.
Stfu
@@julu2731 it's because I liked that game called king of thieves very much.
Yeh this question was for CBSE board eaxm 😂😂 I wonder why he even categorised this question for JEE prep 😂 like what a mockery this is
Come on man I did it in my 10th grade, and I am from the US. Pretty sad you only started it in your 12th.
These type of questions come in jee advanced exm which have to be done in 3 to 4 min and this was an easy one.
Keep up the good work
easy if you know the trick.
I actually solved it by dividing the numerator & denominator by (cosx)^2. Btw, this question was asked in our weekend exam. I'm taking the IIT-JEE exam on May-19-2019.
How did it go?
How did you simplify
ua-cam.com/video/y_XwQkchwrE/v-deo.html
are you alive?
@@karteke Failed to clear the chemistry part ;-;
Gosh, I love that moment when I'm riding along with the explanation and suddenly something clicks into place! Awesome video!
This is not even a JEE level question. Even someone like me who is not preparing for JEE can solve this.
JEE questions are much harder than this
I have solved many jee question but this one is tough
@@biswadevmajhi231 for advance, it's moderate level
@@allipse8224 Can you just stop?
@@nathanielhensley4830 It's like the Gaokao or Oxbridge Entrance Exams. Students study specifically for these tests and attend cram schools for years.
@@ichigo449 I know what it is. It's nothing to be proud of.
I've found out a more inquisitive way to solve it.
use harmonic addition theorem on the denominator,and take the argument of the sine function(or cosine,depending on which one you prefer to use) as u,and differentiate.Also you'll have to to an easy partial fraction which immediately follows.
Just try it!
Harmonic addition ?
Is it possible ?
Coefficients in front of trig functions are not constant
Doesn't necessarily have to have constant coefficients.
try to write the denominator as sin(x+arctan(1/x))
then take the argument of this sine function as u and see what happens
How would you even think about that? OMG
Well the denominator looked that much tempting to me to use the harmonic addition PCreeper
That thoerom is not in the syllabus of the exam i think. But if its a good method then good for you!
Since the order of the denominator of ○'/○^n decreases and becomes simpler when integrated, it may work well to use it as the integrating side of a partial integration.
Recipe for crazy integrals:
1. Generate some crazy function
2 differentiate it.
3. Slap an i integral around the derivative .
4. Unleash on students.
Watch the fun.
I was new to your channel, so I asked you to try JEE. But you actually did it way before... You are awesome bruhh
Love the integral and enjoy your way of teaching ! Always a pleasure to watch and listen ! But this intergral should not be in your calculus 2 EXAM but in a homework that could be evaluated . :D
X sqared means a regular square pattern. Xsinx means multiple waves. cosX added means inverted angles or I. Squaring multiple waves and Angeles ninety means absolute values of waves instances. When you divide one by the other and integrate you get circular ripples.
as someone preparing for jee i remember this question as a format of forcing by parts. it was really long but satisfying.
Here is my solution:
wolfram alpha the integral. DONE
Oon Han yay!!!
Not in jee syllabus
@@rrr1304 dude issa joke
Watching you is much better than social media tho
U know about IIT ? WOWW
@@dr.mikelitoris no u
@@deviprasad_bal I think it's 3rd most difficult exam
@@RC-qi6hs no it's 5th...
1st-CCIE
2nd-GATE
3rd-Gaokao
4th-UPSC
5th-IIT-JEE
@@deviprasad_bal Ty for d correction
@@deviprasad_bal lol out of top 5 difficult Exam 3 exam are of india
Have you ever done a video on the Weierstrass substitution method? It's a cool way to simplify lots of trig integrals.
Yes I have.
: )
Yes but he did not finish Euler's substiutions which are closely related to Weierstrass substitution
He did not finish Euler substitution because he showed one but there are three of them
He has also Ostrogradski method for isolation rational part of integral
which may be useful after Weierstrass or Euler substitution
To find the right integration method looks most difficult. But this seems maybe the best here.
thank you man, for you i'll pass my examen
Thanks sir I'm an JEE aspirant. My exam is on 2019.❤Thanks for the video, I'm studying at 3:00 AM this came in my recommendation and helped a lot. Btw which country are you from?
@Sashank Sriram not every asian is chinese bro.😂He can be from Philippines, korea, japan, Vietnam
@@kushagrapandey2466 ...That doesn't mean he's not Chinese though?
I’m only 16, I think we raise every factor inside the parentheses which will then be the indefinite integral of x squared over x squared times the square of sine x plus the square of cosine x. Sin^2(x) + cos^2(x) = 1, so the equation becomes: x^2 / x^2 * 1, any number multiplied by one stays the same. The equation becomes the indefinite integral of x squared over x squared and then the indefinite integral of 1 dx which results in x plus the integration constant.
x + C
We don’t learn about calculus yet, so this could be horribly wrong and stupid.
our teacher did this with 3 methods
Anna Sir
yes! Anna sir!
Sab batana hai is BHARAT ko, xD!
I never thought that this could be done so smartly
I think we can solve it by substitution also.
1. Divide by (cosx)^2 on numerator and denominator.
2. In numerator, yyou get(xsecx)^2 and in denominator you get (tanx +1/x)^2.
3. Now we can see that derivative of tanx is present in numerator.
Well I might be wrong but can anyone please help me solve this further than this.
Yeah give it in your Calc 2 exam and dont forget to post the students' reaction!!
: )
donitzeti
Wonderful method sir! So much easier and "obvious" once you know the solution (if that made sense)
Thanks for the video. I'mma use this method.
My answer is tan ( x-arctan(x)) + c
I hot this by converting the denominator to one cos
And this is the same answer if you use the formula for tan the diffrence between two angles and then convert tan to sin/cos
Great video. Could you please make some videos about teaching some methods instead of explaining problems. That would really help me! Thanks
Another way from 2021 :) Bring the numerator to 1 via dividing by x^2, and compress the expression in the denominator via trigonometry. You should get ( x^2 * cosec^2( tan-1(1/x)+x ) ) / (x^2 + 1). Then you can directly substitute for tan-1(1/x)+x = u, and get the solution. Btw, when I saw this the first time, I didn't see that the denominator had a whole square lol. Lemme know if I missed something, though I hope I haven't.
I'm not gonna lie, I'd probably have missed this question if it was on my calc 2 exam lol
Simplifying the integral was a bit difficult than integrating it.LOL
I felt this was more of a guess and differentiate question rather than integrate. d(u/v)/dx= (v*du-u*dv)/u^2. Now if you'd notice the denominator is in terms of square. Rest of it is just finding v and v' which can be guessed with not too much efforts.
If you want to call it a scam of a solution, its alright but this gets stuff done fast which is required.
Disclaimer: You get to develop the intuition for the method only by practicing hard. So current JEE students should practice more as compared to cheats like this.
For integrating by parts try this
FIS-DFIS
(FIRST*INTEGRAL OF SECOND) MINUS (INTEGRAL(DERIVATIVE OF FIRST*INTEGRAL OF SECOND dx))
I read the description that it was by parts and solved it easily!
I think it would be more easier if you simplify the denominator... And ans=(-1/2){1/(tan2x+sec2x)}+c
Isn't it also possible to simply do a complex partial fraction decomposition? It may not get easier, but the bruteforce method may save time in exams, if one can't find the trick 🤔
i think it would take equally as long 😅
My jee exam is on 9 Jan 2019
We have a similar question in our Maths book(Cengage)
Only diffn is that in the numerator its x2+20
And its even more tricky!
This is so famous and becomes a cakewalk with two substitutions that my teacher pointed out:
x=tan θ
and then (tan θ - θ) = t
you'll get the answer straightway
Yes I would randomly differentiate something and ask somebody to integrate it for me
*It will be more intresting if we add upper limit and lower limit too*
We are taught this by putting the denominator = t and differentiating. And replacing values its more easy that way but i like the way you make it complex and detailed😃
Leaving decisions for your question paper to thousands on the internet probably breaks some kind of teacher rule
Integration: putting the toothpaste back into the tube
Your pronunciation is literally amazing.
It's good w
Question.... I'm from Varanasi India
Can u try the function :
Y=sqr(cos[x])*cos(∞x)+sqr(x*0.00000001)
and
Y=sqr(cos[x])*cos(∞x)+sqr(-x*0.00000001)
This couled be function with I.
Together its like heart
Sahar haim Yaccov funny one
So its not impossible, ISN'T IT?
ua-cam.com/video/y_XwQkchwrE/v-deo.html
Back in the past: 1+1=2
Now: the integral of x^2/(xsin(x)+cos(x))^2
This is a famous question which came from the coaching modules which use to give it is so easy ❤
1st into derivative of 2nd- integral of (Derivative of 1st*integral of second dx)
I'm in 11th , and solved this problem in 2 mins....of course from India.. Nice thought process of urs though...
Good job!
Actually this question is from CBSE board exam class12th in 2012
what a coincidence!
And the big coincidence is that this video is uploaded almost 11 months ago and I usually like to watch PUBG mobile gameplay videos and luckly this video opened in my UA-cam homepage.
I solved this in 2 seconds ohh such a freaking genius aren't I 👑 /s
I'm still getting over the fact how smart the solution was.
Khosa.... X (cosx)
Dammit cosine x
Bro you make it a piece of cake great effort here ♥
Very ingenious. Congratulation.
This was asked in my unit test
i solved it on my own ,despite being a high school student.It felt great
I mean it's meant to be solved by high school students
@@siddharthbhatt7752 i was on class IX dude.I mean it's not a big deal at all but it was one of the hardest problems i solved during the inception of my integral course
@@reaper3.097 ah ok
Have a stupid question perhaps. In order for the under the integral part to be equal when you multiply and divide by cos(x), don’t you need to specify that cos(x) is not equal zero? Otherwise the equation is false.
Sir please tell me integration of x^sinx
Here we can build differential equation from this result
df/dx (xsin(x)+cos(x))-f(x)xcos(x)=x^2
Me: yet another video of BLACKPENREDPEN
Me (After watching 1:00 ) :
Ohh wait it is BLACKPENBLUEPEN
integration "trick" was very clever...It's now in my math "toolbox".
This que repeat in 2020 jee exam 4th September i am very happy at that moment that i saw this video
Sir we can also do this by dividing by cosx to denominator and numerator..??? Am i right as derivative of lower part will create on upper part??
Well that was pretty easy
If we take g(x) = xsinx + cosx Then the function under integration brings to mind the derivative of the quotient.
It's quite easy to figure it out that if we take f(x) = sinx - xcosx then f'(x) = xsinx and f'g - fg' = x^2
and thus integral is just f/g + C
And no, I'm not from India, I'm from Poland ;)
Zdaje się że ta całka pojawiła się w jednym z komentarzy pod innym filmem Stevena (wcześniejszym niż ten) i ja przedstawiłem wtedy sposób liczenia identyczny jak na tym filmiku
Teraz już tego komentarza nie mogę znaleźć ale pamiętam że miałem taki sam pomysł na tę całkę co pokazany na filmie
You made this very complicated but it can be solved within just 2-3miniutes
What source did you use to learn calculus
I remembered this integral appearing in one of the integration competitions in China. The solution just says the answer is the answer because its derivative is equal to the given function.
What's the D-I rule thing ???????? How do we apply that and what exactly is its use ????????
It's just integration by parts, the entire integral revolves around the first step, you don't need to use your head once you figure out the first step
Wow this question came in jee mains 2020 but he uploaded in 2018
Nice I will study from here 😁😁
Hey. .. just do a subsitution x =tanc. And it will be solved without by parts. ... I am preparing for jee .exam .. it's a tricky one . But by this subsitution it's it is easily solved. In denominator it becomes. sin sin + cos cos. And then we are done.....😊
There is a positive and a negative effect if you put this in your calc2 exam the positive is that your viewer count will increase because your students will watch this the negative is that your students will just memorise the answer
With yoy all is posssible, my respect to you, you are a master in math topics
It was asked jee 2020 national entrance exam for engineering exam sir where did you get the problem sir?
Def put it on the exam
prove deriving > (-xcosx+sinx)/(xsinx+cosx)
Right I'm a class 12 CBSE student. I have done this problem preparing for boards. I'm what people would call a "good" student( above 90%). I hated CBSE and it's method of testing. Honestly, I don't feel like there is a point in rushing students in an exam to solve complicated problems. How does it matter whether you solved it in 2 minutes or 5 minutes? The thinking process is really important.
Why did he do minus 1 in integration instead of plus 1? 5:24
You can’t multiply and divide by a trigonometric function or variable according to integration rules only constant can be multiplied and divided
Yes, but in this case he multiplied by the constant 1, it just took the form of cos(x)/cos(x).
That’s same as multiplying and dividing by cosx
@@DamnitManit no
What kinda exam makes you integrate a function like that??? Like a Harvard exam or something???
What is that DF format at 4:17
Once again a great video by blackpenredpen 💘💖 but i have to say that Its not at all hard the same integral was asked in CBSE class 12 compartment maths board exam 2012 . So its a board level integral not even JEE main level. but yes it definitely required some creativity.
(board exam means the final exams given by high school students in India. equivalent to advanced placement courses in us or Cambridge a levels, as levels in uk)
Now you have to change the channel name to black pen red pen blue pen
This is the easy one from class 12th board
It's not from IIT JEE
This is not a jee question this is a question from ncert and came in cbsc .which is very much easy than jee. Mains and advanced
He solved it in 9 minutes but we have only 2 minutes to solve it in exam.