Getting to Cambridge: STEP by STEP!
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Original post by aymanzayedmannan
Could you help me decompose ?
Are you sure you don't mean ?
Original post by Zacken
Are you sure you don't mean ?
Nah that would have been simple using the Zacken method - I ran into this trying to integrate .
Original post by aymanzayedmannan
Nah that would have been simple using the Zacken method - I ran into this trying to integrate .
Ah, okay. It's an ugly one and don't be be surprised at how ugly your answer will be.
But, basically - the trick here is to note that and then partial fraction it up.
Original post by Zacken
Ah, okay. It's an ugly one and don't be be surprised at how ugly your answer will be.
But, basically - the trick here is to note that and then partial fraction it up.
But, basically - the trick here is to note that and then partial fraction it up.
Disgusting
Original post by Zacken
Ah, okay. It's an ugly one and don't be be surprised at how ugly your answer will be.
But, basically - the trick here is to note that and then partial fraction it up.
But, basically - the trick here is to note that and then partial fraction it up.
That is nasty.
I've seen this method somewhere but I thought there must be an easier trick to make it simpler. Thank you either way. 
Original post by aymanzayedmannan
That is nasty. I've seen this method somewhere but I thought there must be an easier trick to make it simpler. Thank you either way.
I've seen this method somewhere but I thought there must be an easier trick to make it simpler. Thank you either way. If you want, I have the sexiest thing ever for integrating that may or may not cause accidental orgasms.
Original post by Zacken
If you want, I have the sexiest thing ever for integrating that may or may not cause accidental orgasms.
do share, Mr 420
Original post by Zacken
Ah, okay. It's an ugly one and don't be be surprised at how ugly your answer will be.
But, basically - the trick here is to note that and then partial fraction it up.
But, basically - the trick here is to note that and then partial fraction it up.
How do you factorise in your head?
Original post by Student403
Disgusting
Original post by aymanzayedmannan
do share, Mr 420
Here's something a little sexier:
Let and , then:
Unparseable latex formula:
\displaystyle[br]\begin{align*}I + J &= \int\left(\sqrt{\tan x} + \sqrt{\cot x}\right) \;\mathrm{d}x \\&= \sqrt{2} \int\frac{\sin x + \cos x}{\sqrt{\sin 2x}} \;\mathrm{d}x \\[5pt]&= \sqrt{2} \int\frac{(\sin x - \cos x)'}{\sqrt{1-(\sin x - \cos x)^2}} \;\mathrm{d}x \\[5pt]&= \sqrt{2} \sin^{-1}(\sin x - \cos x) + \mathcal{C}_1 \tag{1} \\\end{align*}
Unparseable latex formula:
\displaystyle[br]\begin{align*}I - J&= \int\left(\sqrt{\tan x} - \sqrt{\cot x}\right) \;\mathrm{d}x \\&= \sqrt{2} \int\frac{(\sin x - \cos x)}{\sqrt{\sin 2x}} \;\mathrm{d}x \\&= -\sqrt{2} \int\frac{(\sin x + \cos x)'}{\sqrt{(\sin x + \cos x)^2 - 1}} \;\mathrm{d}x \\&= -\sqrt{2} \ln\left|(\sin x + \cos x) + \sqrt{(\sin x + \cos x)^2 - 1}\right| + \mathcal{C}_2 \tag{2} \\\end{align*}
Adding (1) and (2):
Unparseable latex formula:
\displaystyle\begin{equation*} I = \frac{1}{\sqrt{2}} \sin^{-1}(\sin x - \cos x) - \frac{1}{\sqrt{2}} \ln\left|\sin x + \cos x + \sqrt{\sin 2x} \vphantom{x^{x^x}} \right| + \mathcal{C}\end{equation*}
Original post by Insight314
How do you factorise in your head?
I remembered it, it's a fairly useful identity. But if you can't do it in your head, then it's fairly simple to write out and then compare coefficients.
Original post by Zacken
X
Original post by Student403
This is why I love you
@Ethereal World appreciate this
Original post by Zacken
Seriously dude wtaf 1) How long did that LaTeX take and 2) How on earth did you figure out that technique...
Original post by Student403
Seriously dude wtaf 1) How long did that LaTeX take and 2) How on earth did you figure out that technique...
Oh, no. I can't take credit for that technique! I found that somewhere on the internet, there's no way I'm clever enough to come up with that on my lonesome.
LaTeX took quite a bit of time... ngl
Original post by Zacken
...
Original post by Student403
Same response ^
Looks similar to the classic cos x/(sin x + cos x), but on drugs.
Original post by Zacken
I remembered it, it's a fairly useful identity. But if you can't do it in your head, then it's fairly simple to write out and then compare coefficients.
I know that all this comes with experience, but do you have any other useful identities? For example, I know and completing the nth square generalised form - .Also, I know that but I don't know a general form for factorising .
Original post by Zacken
Oh, no. I can't take credit for that technique! I found that somewhere on the internet, there's no way I'm clever enough to come up with that on my lonesome.
LaTeX took quite a bit of time... ngl
LaTeX took quite a bit of time... ngl
You are a machine...
Original post by aymanzayedmannan
Same response ^
Looks similar to the classic cos x/(sin x + cos x), but on drugs.
Looks similar to the classic cos x/(sin x + cos x), but on drugs.
I literally have no words for that
Original post by Insight314
How do you factorise in your head?
Original post by aymanzayedmannan
...
I remembered it, it's a fairly useful identity. But if you can't do it in your head, then it's fairly simple to write out and then compare coefficients.
Just thought about this some more! Came up with this:
Unparseable latex formula:
\displaystyle[br]\begin{equation*} x^4 + 1 = x^4 + 2x^2 + 1 - 2x^2 = (x^2 + 1)^2 - (\sqrt{2}x)^2 = (x^2 + \sqrt{2}x +1)(x^2 - \sqrt{2}x + 1)\end{equation*}
Zacken style by adding zero then difference of two squares.
Original post by Zacken
Here's something a little sexier:
(clever stuff and fancy latex)
(clever stuff and fancy latex)
Get in my bed.
Original post by Zacken
Oh, no. I can't take credit for that technique! I found that somewhere on the internet, there's no way I'm clever enough to come up with that on my lonesome.
LaTeX took quite a bit of time... ngl
LaTeX took quite a bit of time... ngl
These integrals remind me of Cleo's solutions on MathsStackExchange. I am not sure if you know her but she is the biggest troll on StackExchange. She solves some extremely complicated graduate level integrals (afaik, they are gibberish to me) and then just gives the answer without explaining her logic through it.
Here is one example of it: http://math.stackexchange.com/a/563063 (scroll up to see the actual integral)Quick Reply
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