The h Bar

$$\begin{align}
\int\frac{(2t^2-1)}{(t^2-1)\sqrt{2t^2-1}}\,dt &= \int\frac{(2t^2-2)+1}{(t^2-1)\sqrt{2t^2-1}}\,dt\\
&= 2\int\frac{dt}{\sqrt{2t^2-1}}+\int \frac{dt}{(t^2-1)\sqrt{2t^2-1}}
\end{align}$$ I did like this also. Do you know how to integrate the second term of the final result ?

Ramanujan

@DHMO I wish 3Blue1brown make video on it

DHMO

@Ramanujan looks like transposition is not an essence of linear algebra

@anonymous too complicated... any more elegant ones?

anonymous

14

A: Evaluation of $\int\frac{\sqrt{\cos 2x}}{\sin x}\,dx$

\begin{align} \int\frac{\sqrt{\cos 2x}}{\sin x}\ dx&=\int\frac{\sqrt{\cos^2x-\sin^2x}}{\sin x}\ dx\\ &\stackrel{\color{red}{[1]}}=\int\frac{\sqrt{t^4-6t^2+1}}{t^3+t}\ dt\\ &\stackrel{\color{red}{[2]}}=\frac12\int\frac{\sqrt{u^2-6u+1}}{u^2+u}\ du\\ &\stackrel{\color{red}{[3]}}=\int\frac{(y^2-6y+1)...

DHMO

@anonymous next?

anonymous

10:38 AM

@DHMO next question?

DHMO

ya

anonymous

okay

wait

This is actually easy I guess..

I am looking for a tough one

Ramanujan

@anonymous do you have differentiation challenge?(I also want to participate)

DHMO

@anonymous partial fraction?

anonymous

@Ramanujan I do have

@DHMO Anything..try

Ramanujan

10:41 AM

@anonymous OK,next differentiation please

anonymous

@Ramanujan (This is based on Application Of Differentiation)

DHMO

$\displaystyle=\int\frac{(1+x^2)}{(1+x^2+k)(1+x^2-k)}\ \mathrm dx$

$\displaystyle=\frac12\int\left(\frac{1}{1+x^2+k}+\frac{1}{1+x^2-k}\right)\ \mathrm dx$

@anonymous can i stop here?

anonymous

@DHMO I didn't think of this earlier! Great :)

Yesh

we can stop here

next problem now

DHMO

@anonymous oops, I meant $kx$ not $k$

Ramanujan

@anonymous what is "c"?

anonymous

10:46 AM

@Ramanujan A constant

koolman

@Ramanujan lol

@anonymous good reply

anonymous

@DHMO You can try the differentiation problem ? :)

DHMO

@anonymous then what was your approach?

@anonymous nah, I think @Ramanujan is doing it

anonymous

@DHMO I just divided by $x^2$ up and down and wrote the num as a derivative of a part of the denominator

I think that is more elegant :)

But I like your method too

DHMO

@anonymous oh, the kupur whatever method

anonymous

10:50 AM

KUTUR-PUTUR

:D

That is a great method :P

DHMO

lol

anonymous

Here comes the monster!!

It has a really short solution :)

Ramanujan

@anonymous

> line joining point to origin is independent of the position of point on curve

Means?

anonymous

Show that the angle between

Read the full sentence

the tangent at any point 'A' of the curve

and the line joining A to the origin

is independent of the position of A on the curve

@Ramanujan

Got it now?

Ramanujan

No :(

anonymous

11:01 AM

@Ramanujan It is saying that you need to prove that the angle between the two lines is constant.

You know which two lines they are talking about?

Ramanujan

Yes,tangent line and line joining proving and point

anonymous

@JohnRennie Could you please remove my message from the starboard ? That was a slang :P

Ramanujan

Nice speed whoever did

anonymous

Thank you :)

@Ramanujan you got it ?

understood the meaning ?

Ramanujan

@anonymous for point say (c,c) angle between two lines is 1

@anonymous yes,undersood

anonymous

11:06 AM

yeah!

good :)

Ramanujan

Shukria

Another one?(a bit more challenging)

anonymous

@Ramanujan What ?

You finished the sum ?

DHMO

@anonymous I can't see anything...

Ramanujan

@anonymous which sum?

anonymous

@Ramanujan The problem on differentiation

@DHMO Can you see now ?

DHMO

11:08 AM

@anonymous I mean, I can't see any way

Ramanujan

Angle between them is $\dfrac{\pi}{4}$ for any point

DHMO

oh

!!!!

anonymous

@Ramanujan Wrong.

Ramanujan

So what is correct?

anonymous

The answer includes C.

How did you do it ?

Ramanujan

11:10 AM

OK,wait

anonymous

@DHMO Use kutur-putur

:P

DHMO

@anonymous ya

anonymous

@DHMO done ?

DHMO

$u=\sqrt{x^{2}+x^{-2}}-x$ right

anonymous

@DHMO What next ? Doesn't seem helpful

DHMO

11:13 AM

@anonymous hmm...

can't figure out anything

anonymous

Tried kutur putur ?

Divide by $x^5$ up and down

Ramanujan

@anonymous tan θ = -2/C

?

anonymous

@Ramanujan What is theta ?

Ramanujan

@anonymous angle between two lines

1) tangent line

anonymous

@Ramanujan No no. That is okay. I am asking what is the value of

Ramanujan

11:16 AM

2)line joining origin and any random point

anonymous

theta ?

Simplify it to find theta!

Not tan(theta)

You are close

Ramanujan

$θ=Tan^{-1} (-2/C)$

@anonymous

anonymous

Good. Better written as $\tan^{-1}(C/2)-\pi/2$

@Ramanujan Can you show me your solution? :)

Ramanujan

@anonymous C/2 or 2/C ?

anonymous

@Ramanujan C/2 in my form and 2/C in your form

Think

That is a tricky part

18

Q: Proving that $\arctan(x)+\arctan(1/x)=\pm \pi/2$, could this line of reasoning possibly be correct?

I know that two questions have already been asked about this exercise, but what I'm asking here is if this solution, which sounds rather strange to me, could possibly be correct. The problems is as follows: Prove that $\,f(x)=\arctan(x)+\arctan(1/x)= \pi/2\,$ if $\,x>0\,$ and $\,-\pi/2\,$ if ...

DHMO

11:22 AM

$\displaystyle=\int\frac{x^{-5}}{(x^{-4}+1)\sqrt{\sqrt{x^{-4}+1}-1}}\ \mathrm dx$

Ramanujan

Another differentiation problem?

DHMO

@Ramanujan have I asked you to differentiate $x^{x^{x^{.^{.^{.}}}}}$ before?

Ramanujan

@DHMO no,if yes then I didn't remembered

DHMO

then do it :P

anonymous

@DHMO Good boy/girl :)

Now next one!

@Ramanujan Can you show me your solution to the last problem?

@DHMO for you :)

Ramanujan

11:28 AM

@anonymous sorry bro,too. Lazy. To type

anonymous

okay

try the next problem

i gave

DHMO

$\displaystyle=\int\sqrt{\frac{(\sin x - 1.5)^2 - 0.25}{(\sin x + 1.5)^2 - 0.25}} \ \mathrm dx$

anonymous

Nice try

Go on

DHMO

$\displaystyle\sqrt{\frac{1-\sin x}{1+\sin x}} = \frac{1-\sin x}{\cos x}$

anonymous

Tomorrow can we try parabola sums ? @Ramanujan

DHMO

11:38 AM

@anonymous got it

anonymous

@DHMO How? :D

DHMO

$=\dfrac{\cos x}{1+\sin x}$

then you know i know

Ramanujan

@anonymous I am in first year,parabola and equation haven't introduced

anonymous

@DHMO Even I did like that. But I want to know your next step ! I used 2-sin =z^2

DHMO

@Ramanujan are you doing my question?

@anonymous ok later

anonymous

11:41 AM

sure

Ramanujan

@DHMO oh,it's by y=x^y then taking log and differenciating, then taking dy/dx common,sorry I am busy

DHMO

yes

anonymous

@DHMO I too think I need to go now. When will u be free again ? Then we can continue..

DHMO

we will meet again

anonymous

I will be back in few hours then

cya!

Kenshin

1:38 PM

Well done Federer

Bernardo Meurer

1:55 PM

@ACuriousMind @anonymous Halp

anonymous

@BernardoMeurer RadyToHalp ;-)

Bernardo Meurer

How to evaluate $$\lim\frac{((n+1)!)^2}{(2(n+1)!)}\frac{(2n!)}{(n!)^2}$$

DHMO

@BernardoMeurer bom dia

Bernardo Meurer

@DHMO Bom dia

@anonymous That thing

anonymous

n tends to infinity ?

Bernardo Meurer

1:57 PM

Yep

anonymous

Many terms will cancel out but in the denominator is it $(2(n+1))!$ or $2*(n+1)!$ ?

DHMO

@BernardoMeurer usa a fórmula de Stirling?

Fórmula de Stirling

Em matemática, a fórmula de Stirling recebe o nome do matemático James Stirling e estabelece uma aproximação assintótica para o fatorial de um número. Na sua forma mais conhecida, ela se escreve: n ! ∼ 2 π n ( n e ) n {\displaystyle n\,!\sim {\sqrt {2\pi n}}\,\left...

Bernardo Meurer

The latter

DHMO

(e probablemente nao necesito)

Bernardo Meurer

@DHMO Cannot use approximations :/

DHMO

2:00 PM

@BernardoMeurer it is not an approximation

it is an inequality that is exact

Bernardo Meurer

@DHMO Hm? Stirling is a formula for approximating a factorial, no?

DHMO

@BernardoMeurer it also gives you lower bound and upper bound

Bernardo Meurer

@DHMO Hmm

So, I need this limit for the ratio test on a series

All I need is to know whether the limit is greater than 1 or less than 1

anonymous

It is getting simplified to $$\frac{(n+1)(2n)!}{2*n!}$$

The numerator is obviously larger isn't it ?

Bernardo Meurer

Yes

But it doesn't go to 0, according to wolfram it goes to 1/4

Wolfram

anonymous

2:03 PM

According to wolfram the limit is infinity :P wolframalpha.com/input/?i=(n%2B1)(2n)!%2Fn!+limit+as+n-%3Einfty

Bernardo Meurer

@anonymous You must've simplified badly :/

This is one shitty limit

anonymous

Man you told me the question wrong

I asked you about the denominator

It is not 2*(n+1)!

Your question meant $(2(n+1))!$

Bernardo Meurer

I told it to myself wrong too lol

I also was doing it the wrong way

Oops

When using factorials I feel forced to read the equations screaming

TWO N PLUS ONE!

anonymous

Yeah, first simplify it

Then see what you get

DHMO

$=\displaystyle\lim_{n\to\infty}\frac{(n+1)^2}{(2n+1)(2n+2)}$

$=\dfrac14$

anonymous

2:06 PM

@DHMO Looks good

DHMO

$=\displaystyle\int\frac{\sqrt{2-s}}{(1+s)\sqrt{2+s}}\ \mathrm ds$ right

Bernardo Meurer

@DHMO Cool

Got it, thanks

anonymous

How did the integral come here ? :P

DHMO

@anonymous $s=\sin\theta$

anonymous

Just divide by n^2. That'll do :P

You will get 1/4

DHMO

2:09 PM

@anonymous is it right?

anonymous

@DHMO Is that a new question? :O

DHMO

@anonymous no, it's the question a few hours ago

anonymous

Or related to the limit ?

Oh

@DHMO I used 2-s=z^2

DHMO

@anonymous doesn't look right to me

anonymous

@DHMO I don't think s=sin theta will work

I solved it using 2-s=z^2

Ramanujan

2:13 PM

@DHMO I am confused in component vector of a perpendicular to b

:(

DHMO

@Ramanujan do you understand the projection formula

Ramanujan

Yes

DHMO

which question are we talking about?

Ramanujan

Component vector of a perpendicular to b

Formula derivation

DHMO

2D or 3D?

Bernardo Meurer

2:15 PM

What do I compare $$\sum_{n=1}^{\infty}\frac{\sqrt{n^2+1}}{n^3\sqrt{n+5}}$$ to in order to verify convergence?

DHMO

@BernardoMeurer $\dfrac1{n^2}$ should do

Bernardo Meurer

@DHMO That will go to 0? $$\lim\frac{n^2\sqrt{n^2+1}}{n^3\sqrt{n+5}}$$

Ah

no

DHMO

I mean, $\dfrac{\sqrt{n^2+1}}{n^3\sqrt{n+5}} \le \dfrac1{n^2}$

so it converges

Bernardo Meurer

How do you prove that inequality?

DHMO

well, $n^2+1 \le n^3$ and $n+5 > n$

Bernardo Meurer

2:21 PM

@DHMO True, that basically becomes $n^2\sqrt{n^2 + 1} \le n^3\sqrt{n+5}$

Which is trivial to see is true

for positive $n$'s at least, which is the case

DHMO

so we're done

we also put $\dfrac{\pi^2}6$ as an upper bound of the sum

Bernardo Meurer

But say I wanted to do the comparison test

DHMO

that is the comparison test

Bernardo Meurer

No, but I mean in the form of $\lim\frac{a_n}{b_n}\in \mathbb R\setminus\{0\}$

If that's true than $a_n$ and $b_n$ have the same nature or whatever it's called, so they either both converge or diverge

DHMO

in that case

set $a_n = \dfrac{\sqrt{n^2+1}}{n^3\sqrt{n+5}}$

set $b_n=\dfrac{1}{n^2\sqrt{n}}$

Bernardo Meurer

2:24 PM

Yep

DHMO

then do your whatever test

Bernardo Meurer

Lol, whatever test

Yep, the limit is 1

Sweet

John Rennie

3:05 PM

Have we got any Mathematica gurus in the house?

anonymous

3:23 PM

@DHMO Do you know some organic chem? I needed to ask something about catalytic hydrogenation...

DHMO

@anonymous go ahead

anonymous

@DHMO Can H2/Pt mixure reduce ketones to alkanes ?

( Pt is platinum )

DHMO

I know Pt is platinum lol

anonymous

Hehe..Just wanted to be clear

Do you know ?

DHMO

I don't think so...

anonymous

3:26 PM

In my book it is written that aldehydes are reduced to alkanes by H2/Pt

I'm not sure about ketones

DHMO

never heard of that

@anonymous could you quote your book?

@anonymous you see, to reduce aldehyde/keton to alcohol we use hydride...

anonymous

@DHMO Nothing to quote as such. They just wrote the reaction RCOH (H2/Pt) -> RCH3.

But okay I think i found it

In another book

DHMO

interesting

anonymous

H2/Pt can only reduce RCOH to RCH2OH

I checked my other book

I think that was wrong

DHMO

theoretically we can have RCH2OH + H2 -> RCH3 + H2O

anonymous

3:33 PM

H2/Pt can't reduce aldehyde to alkane unless under drastic conditions. Normally it will be alcohol I guess.

I found this butane.chem.illinois.edu/cyerkes/104_s_12/worksheets/… on the net :)

DHMO

is there basis for "H2/Pt can't reduce aldehyde to alkane unless under drastic conditions."?

anonymous

Anyway thanks a lot for the confirmation @DHMO :)

DHMO

@anonymous the thing with chemistry is that just because you don't know a reaction can happen doesn't mean that reaction can't happen

anonymous

@DHMO Well according to my other book it says that under normal conditions it is reduced to alcohols. So I assume the reduction to alkane is possible only under drastic condition if it is possible at all

DHMO

alright

anonymous

3:37 PM

Till date I couldn't find a detailed book on Undergrad Organic Chem :/. Every book seems to be either incomplete or over-detailed :P

I used LG Wade, Morrison Boyd, Clayden and so many others..

DHMO

@anonymous well a complete and detailed book would have to be as thick as your house's height

anonymous

@DHMO I don't think so. I collect all the details from all the books I have and write them in my copy. And the copies do fit in my house :P

DHMO

@anonymous then there's some details that your copy misses :p

anonymous

Approximately the topics I have is same as in LG Wade :P

@DHMO Maybe!

DHMO

@anonymous how do you make propene from 1-bromopropane?

anonymous

3:49 PM

@DHMO Using some strong base probably at high temperature ?

DHMO

@anonymous be specific lol

anonymous

There are many ways. I would go for E2 elimination with NaOH or KOH

DHMO

not specific enough

anonymous

I can't think of anything else at the moment

you suggest

DHMO

solvent? temperature?

solvent is an integral part of the reaction

anonymous

3:55 PM

@DHMO Alcoholic KOH or NaOH

High temperature

DHMO

how high is high

anonymous

I don't know :P I would use a normal Chem lab burner maybe

DHMO

alright

have you learnt ether synthesis?

anonymous

Williamson

yeah

DHMO

how do you make ethyl ethyl ether from ethanol?

anonymous

3:59 PM

Sorry I am a bit busy studying hydrocarbons now

Will get back to this later

DHMO

sure

Mostafa

4:13 PM

This recent visa ban on Iranians affects Maryam Mirzakhani and most probably, Nima Arkani-Hamed :/

ZeroTheHero

Nima is a Canadian and carries a Canadian passport.

Sir Cumference

Morning

Well, afternoon

Mostafa

Yeah but he most probably holds an Iranian passport too.

anonymous

@Mostafa I was very sad on hearing about the ban :(. What is the world coming to!

ZeroTheHero

I doubt it.

Mostafa

4:17 PM

See this:

17

Q: Are dual nationals (non-US citizens) also affected by President Trump's ban on Iran, Iraq, Libya, Somalia, Sudan, Syria, Yemen?

I am travelling to Australia from Canada in May and I have to take connecting flights to get back to Canada. I stop in Auckland and San Fransisco. I am a Canadian citizen born in Canada however because my parents were born in Iran, I also have Iranian citizenship and passport. I will not be tak...