the "wouldn't it be nice" integral of 1/(1+tan(x))
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- Опубликовано: 23 янв 2022
- When we try to integrate 1/(1+tan(x)), which is the same as the integral of cos(x)/(cos(x)+sin(x)), we will see the usual u-substitution technique wouldn't work right away. Let me show you another integration technique and I call it the "wouldn't it be nice" technique. A similar integral would be the "1, 2, 3, 4" integral here • The 1, 2, 3, 4 Integral!
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Actually, I did this 9 years ago! ruclips.net/video/1G7S15DVM0k/видео.html
nice, it is solvable with substitution tanx = t, but this is mutch shorter
Alternatively, sin(x) + cos(x) = root(2) sin(x + pi/4) then u = x + pi/4 and it's easy from there.
that was my first thought as well
@adandap explain how sin(x) + cos(x) = sqrt(2)sin(x+45)
@@baezmarklimuel7871 Sure, sin(x) + cos(x) = sqrt(2) [ sin(x) . 1/sqrt(2) + cos(x) . 1/sqrt(2)] = sqrt(2) [ sin(x) . cos(pi/4) + cos(x) . sin(pi/4)]. The quantity in the square bracket is sin(x + pi/4) because sin(a+b) = sin(a) cos(b) + cos(a) sin(b) and sin(pi/4) = cos(pi/4) = 1/sqrt(2)
@@adandap nice! thx for sharing this
Wouldn't it be nice if I'd ever even think of doing that.
You never fail to give me fun math problems to do! :)
rt
Just pure gold.
Thanks sir for your support.
Cheers 🥂.
Great technique, now I will use it in my term end examinations
An easier integral? integral cos(x) dx is easier
But it's not nice b/c it's basically irrelevant
Integral constant dx is easier
@@gurkiratsingh7tha993 integration of nothing(0) is easier
But you might remember it's value
You can multiple and divide by sec ^2 x and make u = tanx then continue with partial fractions
It is so fun. every time new method, thanks.
Introduce this substitution: x = arctan(u^2). There will be partial fractions but it works out nicely in the end.
Can you investigate Integral of (1+tanh x)/(1+tan x) ? Does it have a solution? And how can one decide if the antiderivative of some expression can be found?
0:59 my immediate reaction was "sure it would be nice if we only had to integrate 1, but that's not gonna happen". Then you wrote exactly that on the board :)
Very ingenious!
but as I know I can do some thing else ,,, first step : u=tan(x) then du=sec^2(x)dx then dx=du/sec^2(x) so we will have integral of du/u+1(sec(x))^2 ,,,, alternatively u^2=tan^2(x) so u^2=sec^2(x)-1 so (u^2)+1=sec^2(x) so the final form will be integral of du/(u+1)((u^2)+1) then solve it by partial fractions if I am wrong please tell me (:
Haha, not me using the t = tan(x/2) substitution without thinking and making this way more complicated than it needs to be
Had my own way and the antiderivative was x/2 + ...ln|cos(2x)|/4 + ln|sec(2x)+tan(2x)|/4 + c. Saw Mr Steve's answer and found that the trig stuff was equivalent to his trig stuff, after some manipulation.
edit: wouldn't it be nice if you were good at finding complicated ways to write 1 or 0. I wish finding creative ways to rewrite the multiplicative and additive identity were my forte.
thank you for erasing the tiny blue mark at the end
Wait! You shaved? I thought this was an old video you uploaded to this channel.
That's pratically born haber cycle applied to integrals... How fun😂😂😂
Very nice trick.
Same qn arrived in our previous exam
I like integrals that aren’t hard they’re ok :)
The work of genius god damn :p
Taking trig; on my way to calc 1, 2, 3
Old good days
When the answer just popped out, I audibly laughed at the brilliant simplicity
Hay
India🇮🇳
I miss your goatee.
Weierstrass sub u=tg(x/2) would work
This is new video?
Did you shave?
….mind blowing….🤷🏻♂️😱😱😱
Nice
Please do integral of sin(x^2).
cos(x^2)*2x
Derivative of the trig function with the chain rule for the x^2 on the inside
@@trayne5151 integral
@@trayne5151 no that's not correct. If you differentiate cos(x^2).2x you will get 2cos(x^2) - 4x^2.sin(x^2)
@@rithwikanand9451 ya tbh
I'm not certain if this will work, but because e^iz = cos(z) + i sin(z), then sin(z) = Im (e^iz)
Therefore, the integral of sin(x^2) dx equals the imaginary part of the integral of e^ix² dx.
A u substitution, maybe -u² = ix², might be able to produce something in terms of erf(x) or erfi(x). Take the imaginary part, add a constant and you might have your answer.
I don't know that this is correct though. Try it.
Bro why can't it be rationalised? 1-tanx then put tanx=t 1-t/1-t^2 then it's easy
new profile picture?
hello
👍🏻👍🏻
wouldn't it be nice if it was just 1/tanx
wouldnt it be nice
if there was no tanx in this integral...
RUclips: Sin visitas y 3 likes 🤦
Couldn't we just write the denominator as square root 2 * the sine of x + pi/4
🙄
This question is from class 12 NCERT (India)🤣
12th class NCERT mathematics part 2.
you speak so fast