the "wouldn't it be nice" integral of 1/(1+tan(x))

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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[@bprpcalculusbasics]

Actually, I did this 9 years ago! kzbin.info/www/bejne/Z3iahGRrebuAZs0

[@cicik57]

nice, it is solvable with substitution tanx = t, but this is mutch shorter

[@rogerkearns8094]

Wouldn't it be nice if I'd ever even think of doing that.

[@adandap]

Alternatively, sin(x) + cos(x) = root(2) sin(x + pi/4) then u = x + pi/4 and it's easy from there.

[@cartermikovich1369]

that was my first thought as well

[@baezmarklimuel7871]

@adandap explain how sin(x) + cos(x) = sqrt(2)sin(x+45)

[@adandap]

@@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)

[@Z7youtube]

@@adandap nice! thx for sharing this

[@alberteinstein3612]

You never fail to give me fun math problems to do! :)

[@ta_helado]

rt

[@sir.dinosuras4846]

Just pure gold. Thanks sir for your support. Cheers 🥂.

[@sourabhdubey7518]

Great technique, now I will use it in my term end examinations

[@SeeTv.]

An easier integral? integral cos(x) dx is easier

[@not_vinkami]

But it's not nice b/c it's basically irrelevant

[@gurkiratsingh7tha993]

Integral constant dx is easier

[@AirshipToday]

@@gurkiratsingh7tha993 integration of nothing(0) is easier

[@hanumangupta607]

But you might remember it's value

[@abdullahimohamedalasow7783]

It is so fun. every time new method, thanks.

[@user-pq5pm4fz3m]

You can multiple and divide by sec ^2 x and make u = tanx then continue with partial fractions

[@hetsmiecht1029]

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 :)

[@calculus280]

Introduce this substitution: x = arctan(u^2). There will be partial fractions but it works out nicely in the end.

[@justinpark939]

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.

[@MohammedAli-ig2nu]

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?

[@anshumanagrawal346]

Haha, not me using the t = tan(x/2) substitution without thinking and making this way more complicated than it needs to be

[@FireFly_best_girl]

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 (:

[@Mariosergio61]

Very ingenious!

[@michelecastellani1]

That's pratically born haber cycle applied to integrals... How fun😂😂😂

[@beatrixwashere]

thank you for erasing the tiny blue mark at the end

[@kushaldey3003]

Very nice trick.

[@littlelilly7480]

Same qn arrived in our previous exam

[@gddanielk8491]

I like integrals that aren’t hard they’re ok :)

[@alextaunton3099]

When the answer just popped out, I audibly laughed at the brilliant simplicity

[@ahmaurya6187]

Hay

[@ahmaurya6187]

India🇮🇳

[@mathmancalc7753]

Wait! You shaved? I thought this was an old video you uploaded to this channel.

[@fivestar5855]

The work of genius god damn :p

[@Blend33]

Taking trig; on my way to calc 1, 2, 3

[@petereziagor4604]

Old good days

[@ronaldrosete4086]

I miss your goatee.

[@SimsHacks]

Weierstrass sub u=tg(x/2) would work

[@joaquinlemus43]

….mind blowing….🤷🏻‍♂️😱😱😱

[@medhaabhisheki6088]

Nice

[@yoganandasp2085]

Bro why can't it be rationalised? 1-tanx then put tanx=t 1-t/1-t^2 then it's easy

[@jagula]

This is new video? Did you shave?

[@oktayhan9193]

👍🏻👍🏻

[@loneranger4282]

hello

[@rithwikanand9451]

Please do integral of sin(x^2).

[@trayne5151]

cos(x^2)*2x Derivative of the trig function with the chain rule for the x^2 on the inside

[@deeznuts4333]

@@trayne5151 integral

[@rithwikanand9451]

@@trayne5151 no that's not correct. If you differentiate cos(x^2).2x you will get 2cos(x^2) - 4x^2.sin(x^2)

[@Ayush-yj5qv]

@@rithwikanand9451 ya tbh

[@xinpingdonohoe3978]

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.

[@miscccc]

new profile picture?

[@mcig98]

wouldn't it be nice if it was just 1/tanx

[@bobtivnan]

Couldn't we just write the denominator as square root 2 * the sine of x + pi/4

[@aMartianSpy]

wouldnt it be nice if there was no tanx in this integral...

[@yukatiesa]

KZbin: Sin visitas y 3 likes 🤦

[@_ARCATEC_]

🙄

[@anandpandey2112]

This question is from class 12 NCERT (India)🤣

[@gyattrizzV]

you speak so fast

[@sandeepchauhan7596]

12th class NCERT mathematics part 2.