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    \int{\sqrt{tan(x)}}dx
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    (Original post by GeologyMaths)
    \int{\sqrt{tan(x)}}dx
    Is there an antiderivative?
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    (Original post by Student403)
    Very shocking considering he got selected for Trinity Camp from BMO2
    Not yet m8...
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    (Original post by Renzhi10122)
    Not yet m8...
    Wait rly so what's left?
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    (Original post by GeologyMaths)
    \int{\sqrt{tan(x)}}dx
    I know you're a troll, but whatevs, let's dance. Use \tan x instead of tan x.

    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*}

    Obviously not my own work.
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    (Original post by Student403)
    Wait rly so what's left?
    We're all waiting for UKMT to email us. There's a prize announcement that has yet to come out as well, but should either be out soon (it's been far long already!). I think he's also in a slightly bad position, given that 10 other people got the same score as him, and he's on a gap year and hasn't had previous training before, but we'll see soon.
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    (Original post by Renzhi10122)
    We're all waiting for UKMT to email us. There's a prize announcement that has yet to come out as well, but should either be out soon (it's been far long already!). I think he's also in a slightly bad position, given that 10 other people got the same score as him, and he's on a gap year and hasn't had previous training before, but we'll see soon.
    Congratulations on Egmo again, by the way!
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    (Original post by Kvothe the arcane)
    Is there an antiderivative?
    Yep. :-)
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    (Original post by Zacken)
    x.
    I seen a similar method to this in a STEP 1 integral (STEP 1 2002 maybe?), is there a special name for this method? (I + J and I - J and adding them)?
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    (Original post by Zacken)
    I know you're a troll, but whatevs, let's dance. Use \tan x instead of tan x.

    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*}

    Obviously not my own work.
    Ofcourse it isnt your work
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    (Original post by Renzhi10122)
    We're all waiting for UKMT to email us. There's a prize announcement that has yet to come out as well, but should either be out soon (it's been far long already!). I think he's also in a slightly bad position, given that 10 other people got the same score as him, and he's on a gap year and hasn't had previous training before, but we'll see soon.
    Ah ok


    (Original post by GeologyMaths)
    Ofcourse it isnt your work
    Lol
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    (Original post by DylanJ42)
    I seen a similar method to this in a STEP 1 integral (STEP 1 2002 maybe?), is there a special name for this method? (I + J and I - J and adding them)?
    Yep, it works with integrating \frac{\sin x}{\sin x + \cos x} and \frac{\cos x}{\sin x + \cos x}. There isn't a specific name for it because it's very very rarely useful, I think I've only ever seen one other integral amenable to it. An yeah, it is STEP I, 2002, Q7. :-)
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    (Original post by GeologyMaths)
    Ofcourse it isnt your work
    what difference does it make?
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    (Original post by TeeEm)
    what difference does it make?
    The world
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    (Original post by Zacken)
    Yep, it works with integrating \frac{\sin x}{\sin x + \cos x} and \frac{\cos x}{\sin x + \cos x}. There isn't a specific name for it because it's very very rarely useful, I think I've only ever seen one other integral amenable to it. An yeah, it is STEP I, 2002, Q7. :-)
    A shame, it's extremely simple but so so clever
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    (Original post by Zacken)
    Yep. :-)
    Nice solution. Not intuitive to me at all.
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    (Original post by DylanJ42)
    A shame, it's extremely simple but so so clever
    Yep! I have to agree. If you want, Siklos discusses it briefly in Q20 of the Advanced Problem in Core Mathematics booklet. :-) (page 56 for me)

    (Original post by Kvothe the arcane)
    Nice solution. Not intuitive to me at all.
    Thanks! Not my solution though.
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    (Original post by Student403)
    No way????

    physicsmaths is this true??

    Tbf GCSE is crap some people in my class who got A*s actually failed AS Y12 lol
    Unfortunately yes, it is true haha. I was very bad in school I got expelled(permanent) out of my first school within months and didnt give two shits. Was on a D revised a few days for maths and got an A then I knew I was half decent. Hence my teachers carried on putting me down and I showed those pricks lol.
    A in gcse maths-> high score BmO2. I hope I have redeemed myself.
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    (Original post by Renzhi10122)
    Not yet m8...
    I hope 3-4 of those are not eligible for IMO who got >=20 I think a few of those girls arent since they werent selected for egmo.
    Oh well shud find out today right.
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    (Original post by physicsmaths)
    Unfortunately yes, it is true haha. I was very bad in school I got expelled(permanent) out of my first school within months and didnt give two shits. Was on a D revised a few days for maths and got an A then I knew I was half decent. Hence my teachers carried on putting me down and I showed those pricks lol.
    A in gcse maths-> high score BmO2. I hope I have redeemed myself.
    Pretty inspiring
 
 
 
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