You sir are awesome. At first i was bad at maths in school and thought i would never understand advanced mathematics. But you and blackpenredpen inspired me to believe in maths and to have fun with it. After that i easily got A's in exams without learning much and even my teacher was impressed. Thank you for all your work :)
Since cosh(x) - sinh(x) = exp(-x), this "first way" is literally the first way of the last four. To nitpick a bit more, the last method is risky since the formula you start out with only holds for |x| < 1. Interesting that a restricted way yields an unrestricted result.
14:33 I think this formula only works when the absolute value of the argument is strictly less than one ; so you are restricting your primitive to ]-inf,0[
Der Justus You can interchange because of the linearity of the integral, you know, i. e. the integral of a sum of functions is equal to the sum of the integrals of each function. You have done that before yourself (I think), here he only uses the fancy sum sign.
I pondered and I'd have argued that by the Lebesgue-Konvergence theorem (or dominance konvergence theorem) one can interchange the limit and the integral sign since (-1)^n*e^nx is measurable since its continous, even though I am not even sure with this argument and if all conditions are fulfilled. I'd like to hear Flammables opinion or thought process behind what he stated.
Yea, no question it would be too much for a video. But still, as it's mathematics, those steps should be clea, arguable and understood. Anyway, which semester are you in atm, flammable? Also, I may summon BPRP's help for our concern, haha^^
At 3:42 would it be easier to use inspection - the top is the negative differential of the bottom, so it just integrates to -ln(1+coshx - sinhx) right?
welcome back to another segment offff: daddy teach's se.. math at 2:50 can you multiply by one outside the dx that can make my life alot easier if you can i always erase the dx and put it outside...
At the beginning u could also just multiple the top and bottom by e^(-x) and than let u = 1 + e^(-x) so that you don't have to split it in an even and odd function
I wonder you never used the following formula to make your solutions simple. ∫(derivative of f(x))/(f(x)) dx = ln (f(x)) + c If you have a function in the denominator and its derivative is in the numerator, you only need to take log of the function for integration. For example in you solution ∫cosø/sinø dø = ln (sinø) + C you do not have to make substitution. Any way 4 solutions of this problems are wonderful.
Ma boi, could you recommend me some books on calculus+mathematical physics that you currently use and that are in english? I'm studying ME, but am really interested in physics and mathematics and it is very likely that I'll end up studying both later after I'm done with ME. Just for the reference, last "serious" mathematics course I finished in my study was Infinite Series(incl. Fourier and Taylor), Diff. Eqs., Laplace Transforms and basics of Mathematical Physics. And I'm doing statistics(basic) right now, so not exactly much is going on. Thanks!
Lmao why don't you just add and subtract e^x in numerator : 1 = (1+ e^x) - e^x So you get : The integral of 1.dx minus the integral of e^x over (e^x +1) dx... So : x - ln(1+e^x)
ok, I know. By the way, why are you so interested in solving these integrations?I mean I know it's interesting to use different methods to solve these problems,but does these technics or ideas really help you in your everyday work?
This method is too complicated. dx/(1+exp(x)) = exp(-x) dx/(1+exp(-x)) = -d(exp(-x))/(1+exp(-x)), so integral result is -In(1+exp(-x))= x - ln(1+exp(x))
Watching a cute boy doing cute maths, great way to wake up on a Wednesday morning.
You sir are awesome. At first i was bad at maths in school and thought i would never understand advanced mathematics. But you and blackpenredpen inspired me to believe in maths and to have fun with it. After that i easily got A's in exams without learning much and even my teacher was impressed. Thank you for all your work :)
mip mip yay!!!
mip mip it happened the same to me, now maths are fun for me :]
サkelov yey :)
@8:53 , that was a super cringy but funny move there XD
Since cosh(x) - sinh(x) = exp(-x), this "first way" is literally the first way of the last four.
To nitpick a bit more, the last method is risky since the formula you start out with only holds for |x| < 1.
Interesting that a restricted way yields an unrestricted result.
14:33 I think this formula only works when the absolute value of the argument is strictly less than one ; so you are restricting your primitive to ]-inf,0[
That geometric series only works for -1
15:17 why can we interchange it? which argument would you give? I always struggle with those
Der Justus
You can interchange because of the linearity of the integral, you know, i. e. the integral of a sum of functions is equal to the sum of the integrals of each function.
You have done that before yourself (I think), here he only uses the fancy sum sign.
No, that's not right respectively not enough. The limitprocess involved in the infinite sum ruins your argument.
I pondered and I'd have argued that by the Lebesgue-Konvergence theorem (or dominance konvergence theorem) one can interchange the limit and the integral sign since (-1)^n*e^nx is measurable since its continous, even though I am not even sure with this argument and if all conditions are fulfilled. I'd like to hear Flammables opinion or thought process behind what he stated.
Yea, no question it would be too much for a video. But still, as it's mathematics, those steps should be clea, arguable and understood. Anyway, which semester are you in atm, flammable?
Also, I may summon BPRP's help for our concern, haha^^
Der Justus
Please excuse my inappropriate response.
Thanks for reminding me of the limit in the infinite sum.
Hey can anyone explain the derivation of formula for 1/(1-x) in the last process? Ir any derivation of the Formulas link?
At 3:42 would it be easier to use inspection - the top is the negative differential of the bottom, so it just integrates to -ln(1+coshx - sinhx) right?
Fair!
Question: Isn't (1-x)^-1 and therefore sum(x^n ; n=[0,inf)) defined for |x|0 . so the integral is defined only for x
please do 1 week of putnam exam integrals hehe
welcome back to another segment offff:
daddy teach's se.. math
at 2:50 can you multiply by one outside the dx that can make my life alot easier if you can i always erase the dx and put it outside...
At the beginning u could also just multiple the top and bottom by e^(-x) and than let u = 1 + e^(-x) so that you don't have to split it in an even and odd function
I wonder you never used the following formula to make your solutions simple.
∫(derivative of f(x))/(f(x)) dx = ln (f(x)) + c
If you have a function in the denominator and its derivative is in the numerator, you only need to take log of the function for integration.
For example in you solution ∫cosø/sinø dø = ln (sinø) + C you do not have to make substitution.
Any way 4 solutions of this problems are wonderful.
Ma boi, could you recommend me some books on calculus+mathematical physics that you currently use and that are in english? I'm studying ME, but am really interested in physics and mathematics and it is very likely that I'll end up studying both later after I'm done with ME. Just for the reference, last "serious" mathematics course I finished in my study was Infinite Series(incl. Fourier and Taylor), Diff. Eqs., Laplace Transforms and basics of Mathematical Physics. And I'm doing statistics(basic) right now, so not exactly much is going on. Thanks!
To sophisticated. Just 1/(e^x+1)=1-e^x/(e^x+1), and integrating e^x/(e^x+1) is trivial.
Sorry, I didn't watch the video on first four solutions...
why didn't you just integrate cos/sin as cot (x) in the second method?
Awesome video. Another method would be to let x=ln(u) and so dx=1/u and then use PFD. That's the way I've always solved this integral.
Can you try this please. ..
Limit.as n→ infinity of [ dx/(1+x²)ⁿ] the integral takes bonds from 0 to 1
Can you please solve integral from 0 to pi/2 of x^2 cos^n(x) dx please🙏🙏
how integrate x³/e^x+1??
1=1+exp(x)-exp(x). So, we have integral(1)-integral(exp(x)/(exp(x)+1))=x-ln(exp(x)+1)+C
Beautiful :).
Geometric series converges only for abs(q) < 1 then i doubt that last way will be acceptable
haha after trying this myself this feels very interesting
ı felt very differented after this vid :d
i play on 2x speed
boi this dφs my intuition!
Brilliant tricks
How you assume e^x
Is it possible to integrate 1/(1+lnx)
Flammable Maths Even with an approximation?
You can absolutely find a taylors series and integrate that for an approximation, but I believe it will only be valid for 0
Ei - exponential integral
Bro Solve iitjee advanced (entrance exam to an Indian college) questions. They are vey tough and awesome !
Lmao why don't you just add and subtract e^x in numerator : 1 = (1+ e^x) - e^x
So you get : The integral of 1.dx minus the integral of e^x over (e^x +1) dx... So : x - ln(1+e^x)
Now that's a great video! to lift up my day
I never asked, where are you from?
Please 8 ways in 1/1+cos(x)
Heres one way, substitute t = tan(x/2)
Good ideal
Please solve the Problems of IIT JEE Advanced ( Entrance Exam for IITs at undergraduate level in India).
Excellent!
cosh and sinh: integral days
Awesome
Divide numerator and denominator by e^x
intro song please?
it doesn't sound like it
link?
I found it. Thank you! Great videos by the way
why not let u=e^x+1 ，it's much easier.
alright then...I like your videos:)
ok, I know. By the way, why are you so interested in solving these integrations?I mean I know it's interesting to use different methods to solve these problems,but does these technics or ideas really help you in your everyday work?
空想天则 Lol, obviously not if you're not working in math field or studying math but that's not the point.
1:16 Looks like dat boi's become a man😏
This method is too complicated. dx/(1+exp(x)) = exp(-x) dx/(1+exp(-x)) = -d(exp(-x))/(1+exp(-x)), so integral result is -In(1+exp(-x))= x - ln(1+exp(x))
Noice!
You are cool.
integrate (ln x)^2018/(x+1) from 1/e to e
SINCH XDDD Are you purposely shunning its proper pronunciation of shine?
Every time you say cinch, a puppy dies.
First?
Is that still relevant? No.
666 views /0\
First
1=3 in F2 ma boi