HSC 2017 MX2 Integration Marathon (archive) (1 Viewer)

calamebe · Oct 28, 2016

Re: HSC 4U Integration Marathon 2017

InteGrand said:
$\noindent Find $\int _{0}^{\frac{\pi}{2}} \frac{\cos x}{\cos x +\sin x}\, \mathrm{d}x$.$

Also this is another method: http://imgur.com/h2IBmvL

Paradoxica · Oct 28, 2016

Re: HSC 4U Integration Marathon 2017

calamebe said:
http://imgur.com/a/LQn6t

Checked it on wolfram so it's all good.

that is not in it's simplest form.

calamebe · Oct 28, 2016

Re: HSC 4U Integration Marathon 2017

InteGrand said:
$\noindent Let $\lambda , t > 0$. Show that $\int_{0}^{t} \frac{\lambda^{n}}{(n-1)!}\tau^{n-1} e^{-\lambda \tau}\, \mathrm{d}\tau = 1 - \sum_{k=0}^{n-1} e^{-\lambda t}\frac{\left(\lambda t\right)^{k}}{k!}$, for $n=1,2,3,\ldots$.$

http://imgur.com/d7oqSIR

calamebe · Oct 28, 2016

Re: HSC 4U Integration Marathon 2017

Also sorry if my handwriting poses any issues

calamebe · Oct 28, 2016

Re: HSC 4U Integration Marathon 2017

Paradoxica said:
that is not in it's simplest form.

I know, but it's the right answer and I can't be bothered to put it in it's simplest form.

Edit: stuff it ln(3^(5/4)/2^(23/12))

leehuan · Oct 28, 2016

Re: HSC 4U Integration Marathon 2017

InteGrand said:
$\noindent Let $\lambda , t > 0$. Show that $\int_{0}^{t} \frac{\lambda^{n}}{(n-1)!}\tau^{n-1} e^{-\lambda \tau}\, \mathrm{d}\tau = 1 - \sum_{k=0}^{n-1} e^{-\lambda t}\frac{\left(\lambda t\right)^{k}}{k!}$, for $n=1,2,3,\ldots$.$

Wait what... is there something special about the Poisson distribution I should know

Paradoxica · Oct 28, 2016

Re: HSC 4U Integration Marathon 2017

leehuan said:
Wait what... is there something special about the Poisson distribution I should know

Leehuan this isn't the extracurricular marathon take your stats elsewhere yo

leehuan · Oct 28, 2016

Re: HSC 4U Integration Marathon 2017

Meh doesn't work anyway, cause if it were Poisson the upper limit of the sum should be inf

InteGrand · Oct 28, 2016

Re: HSC 4U Integration Marathon 2017

leehuan said:
Wait what... is there something special about the Poisson distribution I should know

The LHS integrates the PDF of a Gamma(n,beta) r.v. (where beta = 1/lambda) from 0 to t and the RHS is the probability a Poisson(lambda*t) r.v. is bigger than or equal to n. The identity thus shows that in a Poisson process, the number of events occurring in time 0 to t when the rate is lambda is Poisson (lambda*t) if we assume the time between events is iid exponential(lambda) (noting that a sum of n iid exponential(lambda) is Gamma(n, (1/lambda)).

InteGrand · Oct 29, 2016

Re: HSC 4U Integration Marathon 2017

Paradoxica said:
don't scare off the children

Find

$\int \frac { x^{ \frac { 2 }{ 3 } } }{ \left( 1+x^{ \frac { 3 }{ 5 } } \right) ^{ 2 } }\, \mathrm{d}x$ .

(Courtesy of leehuan here: http://community.boredofstudies.org...2016-integration-marathon-41.html#post7183192 )

leehuan · Oct 29, 2016

Re: HSC 4U Integration Marathon 2017

InteGrand said:
Find

$\int \frac { x^{ \frac { 2 }{ 3 } } }{ \left( 1+x^{ \frac { 3 }{ 5 } } \right) ^{ 2 } }\, \mathrm{d}x$ .

(Courtesy of leehuan here: http://community.boredofstudies.org...2016-integration-marathon-41.html#post7183192 )

Oh gd.

InteGrand · Nov 1, 2016

Re: HSC 4U Integration Marathon 2017

$\noindent Let $a$ be a real constant with $|a| \neq 1$. Find $\int \frac{1-a^{2}}{1-2a\cos x + a^{2}}\, \mathrm{d}x$.$

jathu123 · Nov 3, 2016

Re: HSC 4U Integration Marathon 2017

InteGrand said:
$\noindent Let $a$ be a real constant with $|a| \neq 1$. Find $\int \frac{1-a^{2}}{1-2a\cos x + a^{2}}\, \mathrm{d}x$.$

$\noindent$I= \int \frac{1-a^2}{1-2a \cos x + a^2}$ d$x \\ = (1-a^2)\int\frac{1}{1+a^2-2a(2\cos^2(\frac{x}{2})-1)} $ d$x \\(1-a^2) \int \frac{1}{(1+a)^2-4a \cos^2 \frac{x}{2}}$ d$x\\\\ $Let $t=\tan \frac{x}{2}\Rightarrow $ d$t=\frac{1}{2}\sec ^2\frac{x}{2}$ d$x\\\therefore $ d$x=2 \cos^2 \frac{x}{2}$ d$t = \frac{2}{1+t^2}$ d$t \\\\ I=2(1-a^2)\int \frac{1}{(1+a)^2-\frac{4a}{1+t^2}}\cdot \frac{1}{1+t^2}$ d$t \\ = 2(1-a^2)\int \frac{1}{(1-a)^2+t^2(1+a)^2}$ d$t \\ = 2(1-a^2)\left (\frac{1}{1-a^2} \tan^{-1}\frac{t(1+a)}{1-a} \right )+C \\ \\ = 2\tan^{-1}\left ( \frac{1+a}{1-a} \tan \frac{x}{2} \right ) + $ C$

pikachu975 · Nov 3, 2016

Re: HSC 4U Integration Marathon 2017

jathu123 said:
$\noindent$I= \int \frac{1-a^2}{1-2a \cos x + a^2}$ d$x \\ = (1-a^2)\int\frac{1}{1+a^2-2a(2\cos^2(\frac{x}{2})-1)} $ d$x \\(1-a^2) \int \frac{1}{(1+a)^2-4a \cos^2 \frac{x}{2}}$ d$x\\\\ $Let $t=\tan \frac{x}{2}\Rightarrow $ d$t=\frac{1}{2}\sec ^2\frac{x}{2}$ d$x\\\therefore $ d$x=2 \cos^2 \frac{x}{2}$ d$t = \frac{2}{1+t^2}$ d$t \\\\ I=2(1-a^2)\int \frac{1}{(1+a)^2-\frac{4a}{1+t^2}}\cdot \frac{1}{1+t^2}$ d$t \\ = 2(1-a^2)\int \frac{1}{(1-a)^2+t^2(1+a)^2}$ d$t \\ = 2(1-a^2)\left (\frac{1}{1-a^2} \tan^{-1}\frac{t(1+a)}{1-a} \right )+C \\ \\ = 2\tan^{-1}\left ( \frac{1+a}{1-a} \tan \frac{x}{2} \right ) + $ C$

How do you already know 4u integration wtf

eyeseeyou · Nov 3, 2016

Re: HSC 4U Integration Marathon 2017

pikachu975 said:
How do you already know 4u integration wtf

He probably accelerated ahead of himself, how else would he get up to 4U integration?

Drsoccerball · Nov 3, 2016

Re: HSC 4U Integration Marathon 2017

pikachu975 said:
How do you already know 4u integration wtf

Paradoxica 2.0.

trecex1 · Nov 3, 2016

Re: HSC 4U Integration Marathon 2017

eyeseeyou said:
He probably accelerated ahead of himself, how else would he get up to 4U integration?

Not that easy to self learn the 4u course, unless you have a lot of spare time and really like maths.

jathu123 · Nov 3, 2016

Re: HSC 4U Integration Marathon 2017

Drsoccerball said:
Paradoxica 2.0.

lmao no way, I shouldn't be even compared to him

jathu123 · Nov 3, 2016

Re: HSC 4U Integration Marathon 2017

pikachu975 said:
How do you already know 4u integration wtf

started ages ago, from watching vids and looking at other peoples working out on marathons. (thats partially how i grasped integration by substitution lol, by looking at people making dx the subject etc). Then participating in marathons strengthens the understanding as well as helping learn new stuff.

eyeseeyou · Nov 3, 2016

Re: HSC 4U Integration Marathon 2017

jathu123 said:
started ages ago, from watching vids and looking at other peoples working out on marathons. (thats partially how i grasped integration by substitution lol, by looking at people making dx the subject etc). Then participating in marathons strengthens the understanding as well as helping learn new stuff.

What other topics are you familiar with?

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