it is called as king property because it fights integral like a king.

Right after surgery, battling cancer and watching blackpenredpen math videos 😂😂😂 ...btw no worries, my chances are at 85 - 90% and one of two tumors has gone today 💪

Its called the "King" because virtually any trig function can be demolished using the substitution. This is especially true when within logs or when you have a sin/cos pairing. It is also fairly easy to change the bounds from pi/2 to another range by using the R-alpha method or by symmetry on a graph

look, whenever I see a definite integral, especially involving trig, I get nervous.

It always the most intimidating integrals that have an elegant solution! :)

I can't wait to surprise my math teacher by using this at the exam

I love maths. This video was so crazy. Especially I+I = 2I

Hi Am shedrach from Nigeria Am perplexed with ya teaching As your channel had become my favourite companion and you my favorite teacher Thanks sire

Absolutely very much important strategy to solve definite integrals. I can assure you that , its my one of the most cherished birthday presents. Thanks Steve. :) :)

I took two full years of calculus and don't recall ever hearing of the King Property. I've had more math fun watching your vids than I ever did in class.

It's not nice to say FU 😂😂😂😂😂!!!!

Sir. You have a super gift for making maths so easy and fun

I watched your recent video using the king property which brought me to here for the proof of the king property. Amazing how much you have improved over time! You were still a great teacher here, but you are much more comfortable in your most recent videos and your English has definitely improved. Thanks for all the great content!

To the person battling cancer: many, many prayers for your COMPLETE HEALING. Please keep on watching this brilliant channel! Stay blessed my friend.

The explanation of this interesting problem can be described in just two words - SIMPLY AWESOME. BPRP - you rock.

Maybe the name is derived from the castling move in chess, where the king and a rook switch places?

I absolutely love your channel!

Very clear explanations of complex problems. Thank you.

It's called King Property because it's used only by Kings of integrals 💪

Wow, so cool! Thanks blackpenredpen.

Oh..my...god, that is brilliant! Thank you so much!!

Nice video!! What a beautiful integral!

Happy New Year, mr Chow! If you celebrate this one.

This Is so beautiful!!!! Thank you!!!!

I realized this property a few weeks ago while solving a random research integral involving logs, I had no idea it was called the King Property. Good to know 😊. Geometrically, the king property just says that whether you start from the left or the right, the area under f(x) should be the same, so reflect the curve along the vertical axis and push the curve to the right appropriately, and the property jumps at you. It’s less random than it seems. Nice video BPRP! 🙌🏽🕺🏽🎊💯✨🤓

I saw integral from this example on the math forum with polish language One of the guys suggested to use substitution to make interval to be symmetric over zero and then use additive property addition of integrands not intervals After using additive property first integral should be integral of odd function on interval symmetric over zero and second should be easy to evaluate

すごい、、、日本の高2ですがただの積分公式とその証明にこれほど驚かされるとは… What a great formula!! I'm surprised that I've been surprised at just a formula and proof (I'm sixteen year-old-student in Japan☺️)

Thank you professor for the lesson

youre a funny dude earned my like today.

Happy New Year

Thank you soooo much

I am touched.

Thank you sir!

lol, I got a Kingsford ad in the process of watching this video.

This is the first putnam integral that I did in my head. I am proud of myself :)

Amazing!

What a clever technique!

Very nice indeed!

Amazing👍👍👍👍

It's amazing! It is to solve an integral indirectly!

If you really want to confuse someone who has not seen this before you could use the i'th root of tan(x)

I love the line about using f of u instead of f U

"it's not nice, say f of u" 😂😂😂 love your sense of humor

What a creative solution, another one for the bag of secret tricks.

Good stuff

amazing

After you do the King property can't you just set x = (pi/2) - x Then that tells you that x = pi/4 Or does that only work because 2*I=1

very cool!!

When you did the previous Putnam integral, after you put the substitution x = tan theta if you would have applied king rule the integral would have been much shorter

thanks a lot

Very nice property!

And here comes the king!

how do you know what i am studying right now ??? 我感到震惊！

Im Japanese and I just started studying English but I could understand this property easily! thank you!

awesome!

1/(1+tan^2½x) =sin^2½x/(sin^2½x+cos^2½x) we use the property ..f(x)/f(x)+f(a+b-x)[integration from a to b]=b-a/2 get the value is π/4 as cos x =sin(π/2-x)

I just calculated the area under the curve, and it came out to be π/4.

BPRP, I have kind of a burning math question that has been on my mind for almost a year now, but I've struggled to find good explanations for it online. Basically, I would like to know if there is an easy way to find the equations for some parametric curve (say {x(t),x^2(t)}) where the speed (independent of direction) of a particle on said curve must be constant. I have tried setting up the relationship where sqrt(x'(t)^2+y'(t)^2)=1, but I haven't gotten much farther. An explanation for how one could do this, or whether it can be done would be greatly appreciated, even if there is a simple answer.

That’s awesome

In our syllabus my teachers told that as this property is the most useful one so we call it the kings property and if the lower limit becomes 0 then it is called queens property 🤔🤔

Here’s an integral you might want to try solving ((1-u)/u)^x du from 0 to 1. It has a definite value when -1>x>1. The solution can be written in terms of elementary functions. Good luck!

Correct name is called the reflection properties..not King properties.. Nice video

No matter the power tangent is raised to, the answer is pi/4, neat

Can you do a video on how to calculate the sum of a harmonic series? I’m struggling to find the sum of this sequence 1/2+1/3+...+1/2019 any help please?

Plz make videos on complex analysis

Impressive !

why did you not tell me this one day before I had my maths exam yesterday.

This is analogous to a trick you can do with finite sums: Sum from n = a to b of f(n) = Sum from n = a to b of f(a + b - n) It's easy to see why this works if you write it out: f(a) + f(a+1) + ... + f(b -1) + f(b) = f(b) + f(b -1) + ... + f(a+1) + f(a) You can actually prove the integral identity by applying this trick to the Riemann sums :)

my skin is peeling, tearing and reaping. my brain is melting, and my hair is falling

Yeah...I like it..

Hmm. I'm thinking about switching sqrt(2) to y and then writing this integral as function of y. It's constant so its derivative is 0. Then I'd try to get another equality by derivating function under integral over y. Ofc it needs to be proved that it's valid method but it should be doable and maybe it's gonna lead to something interesting? I'll give it a try when I have more time. It was probably done by someone cause if it's famous (though I probably didn't remember this one during my studies as I'm forgetful) then I'm sure someone did that as it seems natural way to get new equalities that may be useful somewhere. I wonder if you can do same trick with half-derivatives as I don't know if you can move half-derivatives under integral. Actually this makes me want to find out more about half-derivatives. ;)

You have a continuity issue since there is an asymptote at pi/2.

When I look at the intro and it says Putnam : nope straight to the solution :P

can you always call an integral I and then adding and subtracting them?

I am in strong belief that f (u) is indeed not polite to say, however, love u is! So, I wanted to say: I just called, to say, I love you (bprp)!!

When you use blue pen even though your channel's name is 'black pen red pen' haha.

This problem literally came in our unit test...

@blackpenredpen can i also ask my doubt ..? If yes then where and how?

Every piece on chess board could have such a property.

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Your face reminds me of my friend

I feel like the answer is super obvious, but there's also a chance it's just some calc identity I never learnt, but why does it equal 1 at 9.30? It's super late at night and looking at it makes my head hurt and I can't figure out what the process is . Maybe in the morning I'll have an idea of what's going on but yeah, anyone able to explain?

Almost i would ask that what if we change the squareroot of 2 after that you said that doesnt matter thanks but why mmm

Blackpen Redpen Bluepen! YaY!

Hi, may I ask how to integrate this if the upper limit of the integral is pi? A whole pi. Thank you.

Bro I really like your videos.... But only problem is ur videos are very long...... (I watch ur video in 2× speed) to save time. ...

Hey everybody, for my Honors English IV class we have to write a research paper on a modern theory in the field we want to study in college. I am planning on majoring in math but I dont have any ideas for topics. It has to be something I can understand enough to write 25 pages about, keep in mind I only have up until trigonometry (I'm in it right now) under my belt. Does anyone have any suggestions?

This ques was less than 30sec for me by same method

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These type of questions (on king's property) are taken in beginner level integration classes in India for students preparing for JEE. P.S.: Queen's property: Definite integral of f(x) w.r.t to x from 0 to 2a equals [definite integral of f(x) w.r.t x from 0 to a]+[definite integral of f(2a-x) w.r.t x from 0 to a], where a is a real number, & this can be proved by writing definite integral of f(x) w.r.t x from 0 to 2a as the sum of definite integral of f(x) w.r.t x from 0 to a & definite integral of f(x) w.r.t x from a to 2a, & then substituting x=2a-t in the definite integral of f(x) w.r.t x, where the upper and the lower limits of x are 2a & a respectively, i.e., the second integral of the sum. In addition to that replace dx by (-dt) and you'll get the desired result, or rather I should say, that's a good place to stop (Michael Penn ref. people😊)! Also, my teacher told our class that queen's is generally used with king's property & even though I've no idea where does king's property gets it's name from, I'm sure the name of queen's property is such due to it's usage with king's!!

Let's solve math problems also did a video on this

thumbs up. 😎

Can you prove in a straight triangle when only the sides are divided by 5 then the height to the rest is an integer? I succedded :) . Example- Stand-3 x Stand-4x Over-5x Height -2.4 x Or S-75 s-100 o-125 H-60

There are three main properties in definite integral so they just gave them the names 1)king property 2) queen property 3) jack property According to my opinion

That was King level

"It's not nice to say FU" BAHAHAHAHBABABAHAHAHAHA LOOOL

No need to know calculus. Just knowing the integral is the area under the curve is enough. The function is symmetric around (pi/4, 1/2) so the area under the curve is (1/2)*(pi/2). (=average height*width).

Hello guys...... I'm not champion in mathematics but I have doubt...on area under the curves..... See if I have an integral of x^3 dx from 1 to -1 ...so it covers the area under the X-axis so we can write it as 2*ingration of x^3dx .....because the area between 0to1 is same as the area between -1to0.....see the graph of x^3 and area should be positive... I gaused.. So the answer should be 1/2...but my professor told me that it was 0...and now I'm getting 😕 on that area..... So can somebody tell me why that is so ? 🙏🤒

Hi! you solve the problem but it ........so

If the red integral is = to the black integral , and both denominators are the same , dose that mean that both numerators are = ? If so tan(x)^root 2 =1 so the denominator of the red integral = 2 . How come the step with 2i equation is needed?