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Getting to Cambridge: STEP by STEP! Watch

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    (Original post by Zacken)
    Here's something a little sexier:

    Let I = \int \sqrt{\tan x} \, \mathrm{d}x and J = \int \sqrt{\cot x} \, \mathrm{d}x, then:

    \displaystyle

\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*}

    \displaystyle

\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): \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*}
    Pretty
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    (Original post by Zacken)
    Yes. :no:
    Ahh, I'm with OCR, so I think it's a bit worse.

    I love Physics, so I try to learn more about the topic that I'm learning by doing extra. It is amazing because IOP have lectures for students and I get so much out of it.
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    (Original post by Insight314)
    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. :rofl: Here is one example of it: http://math.stackexchange.com/a/563063 (scroll up to see the actual integral)
    Her profile states that her medical condition prevents her from posting long answers.
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    (Original post by Insight314)

    "I have a medical condition that makes it very difficult for me to engage in conversations, or post long answers, sorry for that." :lol:
    (Original post by Kvothe the arcane)
    Her profile states that her medical condition prevents her from posting long answers.
    He's aware of that.
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    (Original post by Zacken)
    He's aware of that.
    I see :sad: :emo:
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    (Original post by Duke Glacia)
    yes i did ial. #IALrules
    Hell yea!
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    (Original post by Student403)
    Hell yea!
    *ahem* UK Maths *ahem*

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    (Original post by aymanzayedmannan)
    *ahem* UK Maths *ahem*

    Honestly considering I had finished 5 modules in the first term, I wish I did IAL to get them out of the way :emo:
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    (Original post by Student403)
    Honestly considering I had finished 5 modules in the first term, I wish I did IAL to get them out of the way :emo:
    It's a pain, isn't it? Considering Zacken did all his modules other than FP2 and FP3 already. I wish I got C34 and S2 out of the way just so I wouldn't have to go to school to give those mocks
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    (Original post by aymanzayedmannan)
    It's a pain, isn't it? Considering Zacken did all his modules other than FP2 and FP3 already. I wish I got C34 and S2 out of the way just so I wouldn't have to go to school to give those mocks
    First world problems... When u finishin' maffs too fast for your school :rofl:
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    (Original post by Student403)
    Hell yea!
    having Jan exams is a big benefit bruh. No doubt. Feck michael gove
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    (Original post by Duke Glacia)
    having Jan exams is a big benefit bruh. No doubt. Feck michael gove
    Apparently there's gonna be October exams from now on too for IAL xD
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    (Original post by aymanzayedmannan)
    Apparently there's gonna be October exams from now on too for IAL xD
    FFS so lucky
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    (Original post by aymanzayedmannan)
    It's a pain, isn't it? Considering Zacken did all his modules other than FP2 and FP3 already. I wish I got C34 and S2 out of the way just so I wouldn't have to go to school to give those mocks
    You want to sit C34 and S2 with 125/125 and 70/75 boundaries? You're lucky you're sitting them in June, tbh
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    (Original post by Zacken)
    You want to sit C34 and S2 with 125/125 and 70/75 boundaries? You're lucky you're sitting them in June, tbh
    I mean (I know I might sound unreasonable here) 125/125 is an unpredictable feat. You might be brilliant at C34 but still not manage it simply due to being unlucky (take you as an example), but I think with enough brute force it is possible to get it if you go full out being a tryhard in the exam. The way I approached my easy ones last year (C1, C2, C3) was finish in record time to literally scrutinise every little detail of the questions when checking over and it seemed to work well. I dunno if it works in an exam out of 125, which is much higher than my 75 ones. But :dontknow:
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    (Original post by Student403)
    I mean (I know I might sound unreasonable here) 125/125 is an unpredictable feat. You might be brilliant at C34 but still not manage it simply due to being unlucky (take you as an example), but I think with enough brute force it is possible to get it if you go full out being a tryhard in the exam. The way I approached my easy ones last year (C1, C2, C3) was finish in record time to literally scrutinise every little detail of the questions when checking over and it seemed to work well. I dunno if it works in an exam out of 125, which is much higher than my 75 ones. But :dontknow:
    I agree with you, the exam being out of 125 doesn't make too much of a difference I'd wager, you need more stamina and examination dexterity, but generally not much of a difference. I finished really quickly as well, but then I spent the remaining time eating food and making eye contact with my invigilators to freak them out. :lol:

    But meh, to be honest, full UMS isn't such a big deal. I'm quite the opposite of a perfectionist. :rofl: I think A-Level does a bad job with giving us UMS since there's such a big culture shock that we need to adapt to at uni where getting 100/100 is impossible, etc...
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    (Original post by Zacken)
    I agree with you, the exam being out of 125 doesn't make too much of a difference I'd wager, you need more stamina and examination dexterity, but generally not much of a difference. I finished really quickly as well, but then I spent the remaining time eating food and making eye contact with my invigilators to freak them out. :lol:

    But meh, to be honest, full UMS isn't such a big deal. I'm quite the opposite of a perfectionist. :rofl: I think A-Level does a bad job with giving us UMS since there's such a big culture shock that we need to adapt to at uni where getting 100/100 is impossible, etc...
    Ah that's cool.. but... what? FOOD!??!!? YOU GOT FOOD!?!??!?!!? OMG

    I agree! I don't think it bodes well with uni. Although I think it is a nice habit to pick up - going for the best you can possibly get. Perhaps it isn't possible in the UK but I think for me in the US it'll work out well because 4.0 GPAs are possible, even if hard
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    (Original post by Student403)
    Ah that's cool.. but... what? FOOD!??!!? YOU GOT FOOD!?!??!?!!? OMG

    I agree! I don't think it bodes well with uni. Although I think it is a nice habit to pick up - going for the best you can possibly get. Perhaps it isn't possible in the UK but I think for me in the US it'll work out well because 4.0 GPAs are possible, even if hard
    Yeah, you're allowed to bring in stuff like cereal bars and stuff, so are you!

    Ah, yeah. You've got that in the US. I think in Cambridge Maths, to get a first in second/third year, you need about \approx 9 \% :lol:
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    (Original post by Zacken)
    Yeah, you're allowed to bring in stuff like cereal bars and stuff, so are you!

    Ah, yeah. You've got that in the US. I think in Cambridge Maths, to get a first in second/third year, you need about \approx 9 \% :lol:
    Whaaa? They don't let us because of the wrappers - might have information somehow written on them -.-

    LOL that sounds about right. Imagine what it takes to be senior wrangler... :adore:
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    (Original post by Zacken)
    You want to sit C34 and S2 with 125/125 and 70/75 boundaries? You're lucky you're sitting them in June, tbh
    I know!! I counted my blessings that I didn't sit them after I saw the papers. June C34 papers tend to have super low boundaries as well.

    (Original post by Zacken)
    But meh, to be honest, full UMS isn't such a big deal. I'm quite the opposite of a perfectionist. :rofl: I think A-Level does a bad job with giving us UMS since there's such a big culture shock that we need to adapt to at uni where getting 100/100 is impossible, etc...
    Honestly, everything after 90 UMS is basically the same in terms of raw marks. 90 UMS on one paper could be 100 UMS on another - it simply is a matter of luck. 403 and I were discussing this about the SAT that day - that everything past 2250 is basically the same thing.

    (Original post by Student403)
    I agree! I don't think it bodes well with uni. Although I think it is a nice habit to pick up - going for the best you can possibly get. Perhaps it isn't possible in the UK but I think for me in the US it'll work out well because 4.0 GPAs are possible, even if hard
    Ever heard of the normal distribution, my friend? The bell curve is a wonderful thing in North American unis.
 
 
 
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