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blackpenredpen
Feb 6, 2019

Any video ideas?

If you have any math discovery or fun problem that you want me to present in a video, please leave it here (with solutions hopefully). I will try my best to make it into a video for you and will credit you in the video.

Yrc Murthy
Feb 6, 2019

Sir, I have this new Idea what if you have trigonometric ratios to a base i.e. sin 90 to base π is 1 ? I'm working on it and I named it Trigology. Sir, I want you to help me out.

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blackpenredpen
Feb 9, 2019

Ok! Thanks! I will be interested to take a look.

Yrc Murthy
Mar 8, 2019

@blackpenredpen okay sir, will you make a video on it ?

Mursaleen
Feb 6, 2019

Can you please explain why we add "2kpi" when finding complex roots? And also show the roots on the Argand diagram, giving us a detailed explanation? Thanks.

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Megha Jogithaya
Feb 7, 2019

Hi,

Can you please solve the question in the image?


Elliot Ede (Student-6F)
Feb 7, 2019

Can’t you just solve it graphically? Set theta equal to your equation, which is in terms of theta, in a graphing calculator (unless you are not allowed to use a calculator)

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Megha Jogithaya
Feb 7, 2019

This is a question from a JEE Advanced paper.

Another thing, does this site have a glitch? Because the typed text is shown in wrong order.

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ahaar13
Feb 9, 2019  ·  Edited: Feb 9, 2019

I just wrote up a solution :) Here's a link to the pdf file: https://drive.google.com/file/d/1U3MJz4kklkZjAqbl6pbKg5w0bJ1wqqhr/view?usp=sharing

ahaar13
Feb 8, 2019

I suppose I should have posted my integral here... Anyway, I'd love to see a video about the alternate methods I posted in the discussions for solving the Putnam integral you just posted a video about. If you need more of a worked solution than the one I gave, then I can provide one :)

KING SOBIESKI
Feb 9, 2019

x*sin(x)=k


Find x in terms of k.

michael einhorn
Feb 10, 2019

the roots of this polynomial

https://www.wolframalpha.com/input/?i=x*sin(x)+taylor+series

KING SOBIESKI
Feb 12, 2019

@michael einhorn Not looking for roots. I want to rearrange the equation so that I can isolate x.

ahaar13
Feb 12, 2019

@KING SOBIESKI That's impossible. You're asking to find the inverse, which in this case you can't do. Even if you could, it would not work on the whole real line, it would work in sections of length pi.

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michael einhorn
Feb 10, 2019

Simple Harmonic motion is governed by the differential equation F = -kx or d^2x/dt^2 = -kx/m.

This equation models the oscillation of an object on a spring.

The solution to this equation is a linear combination of sine cosine.

https://www.wolframalpha.com/input/?i=d%5E2x%2Fdt%5E2+%3D+-+kx

How is the integral evaluated, neither my AP physics nor multi variable calc teacher at my high school knew.

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ahaar13
Feb 10, 2019

That is a second order linear differential equation with constant coefficients. This one is easier because there is no x'(t) term, so we can simply realize that both sin and cos are solutions, and they are linearly independent so their linear combination is a general solution (this is a classic theorem in differential equations). More generally to solve things like this, make the substitution x(t)=e^(at) where a is a constant that you will find. Here, you'll find that a is an imaginary number and you can use Euler's formula to turn that into the sin and cos (there are two values of a, and we generally look for real solutions so we throw out the solutions multiplied by i). You can read more about this at http://tutorial.math.lamar.edu/Classes/DE/SecondOrderConcepts.aspx

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Lorenzo Luccioli
Feb 11, 2019

Could you make a video about why in some cases we have to simplify sqrt(x^2) to |x| instead of x and when we have to be more careful about that?

Do I have to be careful every time I simplify by 2 inside an exponent? For example if I’m simplifting x^(5/2)*x^(1/2) to x^3 do I have to put the absolute value?

Thank you for your time : )

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Ibrahim Elsadek
Feb 12, 2019

http://mathserver.neu.edu/~lovett/teaching/math1140/challenge.html

The question is number 3. Using a general cubic function f(x) = ax^3 +bx^2 + cx + d,

find the tangent line to f(x) at x=x0, then find where that tangent line intersects the cubic function, other than at x0 of course.

Here is the link to my solution.

https://docs.google.com/document/d/1_qpDZEkmEnig9GGQQhLBziU_GjzN297AN3J29u4QD54/edit?usp=gmail

Qubix
Feb 15, 2019

The integral of screwdriver, or how to turn English words into integrals


eliottgabrielle74
Feb 21, 2019

Hello ! You should try to determinate the domain of anything raised to anything (x^y). Here is my answer ! https://www.blackpenredpen.com/discussions/_math/domain-of-x-y

@blackpenredpen :D

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Omair Siddique
Feb 23, 2019

please solve this equation


I’m really sorry I’ve got no solutions to it. As I have seen your explanations on YouTube for unique topics, I think you can solve it. Thank you, sir.
It was in NSTC Screening test (Pakistan)

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Alex Gonzalez
Jun 8, 2019

That would be x=3, since after simplifying, you get

(2-x)^2*(x-2)^2=1, and 1*1 is 1.

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harshil bhatia
Feb 23, 2019

Integrate The following

(x^6057 + x^4038 + x^2019)* [(2x^4038 + 3x^2019 + 6)^1/6]

Daunting looking but has a very innovative solution


Alan Tang
Feb 25, 2019

If a^b=b^a ,a and b>0

How to solve it by showing steps😂

KING SOBIESKI
Feb 26, 2019

There are already videos on this:

https://www.youtube.com/watch?v=PI1NeGtJo7s

https://www.youtube.com/watch?v=z65wrFB0W-Y

Math Nerd 1729
Mar 1, 2019

Remember that video you just did about all 3 cases of y' times y'' equalling y'''? Well, here's my solution to the general case 2 [Hopefully this comment pushes through unlike in the YouTube video]:

General Case 2: C2>0

1/(u^2+C2) du = 1/2 dx Integrate both sides: arctan(u/sqrt(C2))/sqrt(C2) = x/2 + C3 Multiply both sides by sqrt(C2): arctan(u/sqrt(C2)) = xsqrt(C2)/2 + C4 Take the tangent of both sides: u/sqrt(C2) = tan(xsqrt(C2)/2 + C4) Multiply by sqrt(C2): u = sqrt(C2) tan(xsqrt(C2)/2 + C4) Since u = dy/dx: dy/dx = sqrt(C2) tan(sqrt(C2)*x/2 + C4) Multiply by dx: dy = sqrt(C2) tan(sqrt(C2)*x/2 + C4) dx Integrate both sides: y = 2 sqrt(C2) Ln|sec[xsqrt(C2)/2 + C4]|/sqrt(C2) + C5 Cancel sqrt(C2) to get all answers from case 2: y = 2 Ln|sec[xsqrt(C2)/2+C4]| + C5

Lo and behold!

Math Nerd 1729
Mar 1, 2019

Oh, and here's case 3:


General Case 3 where C2<0:


Let -C2 = C3

Hence u^2+C2 can and will be written from here onwards as u^2-C3 where C3>0


special mini-case: u^2-C3=0, thus y= + or - sqrt(C3)+ C4 (this is where linear functions come from) [proof is trivial for this mini-case and is left as a mini-exercise for the reader]


Now, if u^2-C3 isn't 0:

1/sqrt(u^2-C3) du = 1/2 dx

Integrate both sides:

-1/sqrt(C3) arctanh(u/sqrt(C3)) = x/2 + C4 [where arctanh(w) is the inverse of tanh(w)]

Multiply both sides by -sqrt(C3):

arctanh(u/sqrt(C3)) = -xsqrt(C3)/2 + C5

Rearrange the right hand side:

arctanh(u/sqrt(C3)) = C5 - xsqrt(C3)/2

Taking the hyperbolic tangent of both sides:

u/sqrt(C3) = tanh(C5 - xsqrt(C3)/2)

Multiply both sides by sqrt(C3)

u = sqrt(C3) tanh(C5 - xsqrt(C3)/2)

Since u = dy/dx

dy/dx = sqrt(C3) tanh(C5 - xsqrt(C3)/2)

Multiply by dx

dy = sqrt(C3) tanh(C5 - xsqrt(C3)/2) dx

Integrate both sides:

y = sqrt(C3) * -2/sqrt(C3) * Ln(cosh(C5 - xsqrt(C3)/2)) + C6

Cancel sqrt(C3):

y = -2 Ln(cosh(C5 - xsqrt(C3)/2)) + C6

We can convert the negative sign outside the ln to taking the reciprocal of the input of ln. Since 1/cosh(w) = sech(w), the final result becomes:

y = 2 Ln(sech(C5 - xsqrt(C3)/2)) + C6

Lo and behold!

phil.petrocelli
Mar 7, 2019

https://www.reddit.com/r/calculus/comments/axqy9p/need_help_with_this_integral/

I have tried this problem in a variety of ways and cannot seem to get it. I have messed around in Desmos and see that the integral is always pi/4, but cannot seem to get the trick, algebraically. I have tried in the complex world, as well, but the nth power seems to cause problems no matter how I try. And since nth powers of sine and cosine behave differently for odd and even powers, it appears to be a mess of an integral combined with infinite series or something. Attaching a pic of the problem, too. Would make a great video, I think.


Thanks for all you do. Best regards from Seattle.


Phil




blackpenredpen
May 1, 2019

Phil, here: https://youtu.be/9xaGYqiOkPM

phil.petrocelli
May 10, 2019

@blackpenredpen OMG, thank you! So many crazy integration techniques that don't seem to be covered in many of the books I have. Any ideas where I might find more things like this? I have a ton of calculus books, and even a few that specialize in integration techniques, and nothing like this is covered. I know Brilliant.org has stuff, but since I joined, the advanced integration piece is still not done and public yet. I am doing a write-up of the solution to this on my math blog at mymathteacheristerrible.com. Thank you so much!

Phil deJong
Mar 11, 2019

Hi BlackPenRedPen! Really enjoying your videos as I navigate Calc 2. I keep coming across what I think are telescoping series problems that I just can't figure out. They look like partial fractions but when I break 'em up I get three terms as opposed to two. Check this out:



Charlie Drinnan
Mar 15, 2019

Really enjoy your videos. Particularly the hard integrals. How about the double integral of




Hint, look at the derivatives of (sin x)^x and (x*cot x+ln sin x)

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kthim_imeri
Mar 16, 2019

Hi BlackPenRedPen. I have here a very interesting integral, which shows that you cannot swap integration and limit (in general):



[Solution Spoilers:] The right-hand-side is infinity, the left-hand-side is the limes for h going to 0 of (log(4h+9)-log(4h+1))/2 , which is log(3).


Extra info: These are known as hyper singular integrals. More infors under Hilbert transform, principal value integral. Also: Hadamard finite part integral.

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sunchyvak6
Mar 27, 2019

Could you limit this series above and belov for an error for 0.01%? Also it is interesting to illustrate how can geometry solve such things


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Eunice Zhang
Mar 28, 2019

i was curious if you can integrate for l'hopitals instead of taking the derivative. would that necessarily work?

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