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

InteGrand · Feb 24, 2017

Re: HSC 2017 MX2 Integration Marathon

Here's a much easier one.

Find $\int \sqrt{x^{2} + 4x + 8}\,\mathrm{d}x$ .

BenHowe · Feb 24, 2017

Re: HSC 2017 MX2 Integration Marathon

InteGrand said:
$\noindent What question? When I say remove 2, something like this will converge: $\prod_{k=0}^{\infty} \cos\left(\frac{t}{2^{k}}\right)$. You can then integrate that (as in it will be integrable). It can be shown that that product converges to $\frac{\sin t \cos t}{t}$. To make the resulting integral easier (expressible in terms of use of elementary functions), you could place a $t$ in front of the product to cancel the $t$ from the denominator of the result of the infinite product $\Bigg{(}$i.e. $t \prod_{k=0}^{\infty}\cos \left( \frac{t}{2^{k}} \right)$$\Bigg{)}$.$

How do I find what it converges to?

InteGrand · Feb 24, 2017

Re: HSC 2017 MX2 Integration Marathon

BenHowe said:
How do find what it converges to?

Here's a hint.

$\noindent You can use induction and double angle formula to show that $\sin x = 2^{N}\sin \left(\frac{x}{2^{N}}\right) \prod_{k=1}^{N} \cos \left(\frac{x}{2^{k}}\right)$ for all positive integers $N$.$

BenHowe · Feb 24, 2017

Re: HSC 2017 MX2 Integration Marathon

InteGrand said:
Here's a much easier one.

Find $\int \sqrt{x^{2} + 4x + 8}\,\mathrm{d}x$ .

Is there a trick for $\int(sec^{3}\theta)d\theta$

InteGrand · Feb 24, 2017

Re: HSC 2017 MX2 Integration Marathon

BenHowe said:
Is there a trick for $\int(sec^{3}\theta)d\theta$

Yeah, if you call it a trick.

BenHowe · Feb 24, 2017

Re: HSC 2017 MX2 Integration Marathon

InteGrand said:
Here's a hint.

$\noindent You can use induction and double angle formula to show that $\sin x = 2^{N}\sin \left(\frac{x}{2^{N}}\right) \prod_{k=1}^{N} \cos \left(\frac{x}{2^{k}}\right)$ for all positive integers $N$.$

I still dont understand how to do it

InteGrand · Feb 24, 2017

Re: HSC 2017 MX2 Integration Marathon

BenHowe said:
Is there a trick for $\int(sec^{3}\theta)d\theta$

See this document: https://www.math.ubc.ca/~feldman/m121/secx.pdf .

InteGrand · Feb 24, 2017

Re: HSC 2017 MX2 Integration Marathon

BenHowe said:
I still dont understand how to do it

$\noindent Also use $\lim_{\theta\to 0}\frac{\sin \theta}{\theta} = 1$.$

InteGrand · Feb 24, 2017

Re: HSC 2017 MX2 Integration Marathon

BenHowe said:
I still dont understand how to do it

$\noindent Call that $N$-th partial product $P_{N}$, so with the hint, we have $P_{N} =\frac{\sin x}{2^{N}\sin \left( \frac{x}{2^{N}}\right)}$. Taking the limit as $N\to \infty$ shows that the infinite product is $\frac{\sin x}{x}$, using the fact that $\lim_{N\to \infty} 2^{N}\sin \left( \frac{x}{2^{N}}\right) = x$.$

BenHowe · Feb 24, 2017

Re: HSC 2017 MX2 Integration Marathon

InteGrand said:
See this document: https://www.math.ubc.ca/~feldman/m121/secx.pdf .

You are legit so good at this stuff and have awesome resources... Wish you were my maths teacher for HSC

Paradoxica · Feb 24, 2017

Re: HSC 2017 MX2 Integration Marathon

InteGrand said:
By inspection

:')

Paradoxica · Feb 24, 2017

Re: HSC 2017 MX2 Integration Marathon

BenHowe said:
Is there a trick for $\int(\sec^{3}\theta)d\theta$

Here's a trick method, I suppose...

$\int \frac{\cos{\theta}}{\cos^4{\theta}} \, \text{d}\theta \stackrel{s = \sin{\theta}}{=} \frac{\text{d}s}{(1-s^2)^2}$

Method one: Exhaustive Partial Fractions. Exercise left to the reader who is bothered.

Method two: Mixed Partial Fractions and a trick substitution.

Observe that $1 = \frac{1}{2}\left(s^2+1+(1-s^2)\right)$$

Replace this into the numerator and split the integral into two parts.

$2\int \sec^3{\theta} \, \text{d}\theta = \int \frac{s^2+1}{(s^2-1)^2} \, \text{d}s + \int \frac{1}{1-s^2}$

Note that for the part that still has the perfect square on the denominator, upon dividing the entire fraction by s², the numerator is the derivative of s-s⁻¹, which is simply the reverse chain rule. The part which is now a simple quadratic is a matter of partial fractions, which is easily done by inspection.

$2\int \sec^3{\theta} \, \text{d}\theta = \int \frac{\text{d}\left(s-\frac{1}{s}\right)}{\left(s-\frac{1}{s}\right)^2} + \frac{1}{2} \int \frac{1}{1+s} + \frac{1}{1-s} \, \text{d}s = -\frac{1}{s-\frac{1}{s}} + \frac{1}{2}\log{\left(\frac{1+s}{1-s}\right)}$

$2\int \sec^3{\theta} \, \text{d}\theta = \frac{\sin{\theta}}{\cos^2{\theta}} + \frac{1}{2} \log{\left( \frac{1+\sin{\theta} }{\cos{\theta}}\right)^2} = \sec{\theta}\tan{\theta} + \log{(\sec{\theta}+\tan{\theta})}$

Constant of integration omitted for brevity

omegadot · Feb 24, 2017

Re: HSC 2017 MX2 Integration Marathon

BenHowe said:
So when should I remember to see if this works (the reverse quotient rule)?

$\noindent You might like to try out the rule on the following question. And just for fun, let us find this integral in three different ways with a few hints provided along the way.$

$\noindent Find $\int \frac{x^2}{(x \sin x + \cos x)^2} \, dx $ as follows$

$\noindent \textsc{Method I}: Use the reverse quotient rule method.$

$$\noindent \textsc{Method II}: Use a substitution of $x = \tan u$ following by a substitution of $t = \tan u - u$

$\noindent \textsc{Method III}: By first observing that $\frac{d}{dx} (x \sin x + \cos x) = x \cos x$, use integration by parts.$

si2136 · Feb 24, 2017

Re: HSC 2017 MX2 Integration Marathon

Wait, so are we allowed to use methods not known in the HSC Syllabus? That would make everything so much easier

Drongoski · Feb 24, 2017

Re: HSC 2017 MX2 Integration Marathon

si2136 said:
Wait, so are we allowed to use methods not known in the HSC Syllabus? That would make everything so much easier

Don't see why not. Not something clandestine, is it?

omegadot · Feb 24, 2017

Re: HSC 2017 MX2 Integration Marathon

$$\noindent While you may not know a particular method explicitly, it is possible in a Question 16 type of question in the HSC you could be \textit{guided} through using a previously unknown technique or method.$

Paradoxica · Feb 24, 2017

Re: HSC 2017 MX2 Integration Marathon

BenHowe said:
$\text{For}\int\frac{xe^{x}}{1+e^{x}}dx\\\text{re-write as}\int x (\frac{e^{x}}{1+e^{x}})dx\\\text{Then do IBP and a substitution in the remaining integral and bob's your uncle}$

That doesn't work, it's still a non-elementary integral.

The answer returned by Mathematica is

xlog(1+e^x) + DiLog(-e^x) +C

The DiLogarithm is non-elementary and can only be simplified under specific circumstances, none of which are met by this integral.

Paradoxica · Feb 24, 2017

Re: HSC 2017 MX2 Integration Marathon

si2136 said:
Wait, so are we allowed to use methods not known in the HSC Syllabus? That would make everything so much easier

As long as it's not dependent on circular assumptions or unproven results (so preferably, has multiple proofs, hopefully some elementary ones), my head teacher told me that I was free to use any theorems I wanted to, that didn't assume the truth of some or all of the question you were going to use it on.

For example, one of the questions I was given technically required the monotone convergence theorem/dominated convergence theorem, in order for the question's proof to be valid, which I cited, and was marked as correct.

Just make sure they can't find any microscopic flaws in your argument, as nobody knows how strict the HSC markers are with extracurricular mathematical reasoning.

si2136 · Feb 24, 2017

Re: HSC 2017 MX2 Integration Marathon

Paradoxica said:
As long as it's not dependent on circular assumptions or unproven results (so preferably, has multiple proofs, hopefully some elementary ones), my head teacher told me that I was free to use any theorems I wanted to, that didn't assume the truth of some or all of the question you were going to use it on.

For example, one of the questions I was given technically required the monotone convergence theorem/dominated convergence theorem, in order for the question's proof to be valid, which I cited, and was marked as correct.

Just make sure they can't find any microscopic flaws in your argument, as nobody knows how strict the HSC markers are with extracurricular mathematical reasoning.

That's why I'm planning to use only known methods in my HSC externals. I asked the head teacher and I'm allowed to use any method as long as they have formal documentation on it and it leads to the answer (For ex. Heaviside Cover Up)

InteGrand · Feb 24, 2017

Re: HSC 2017 MX2 Integration Marathon

si2136 said:
Wait, so are we allowed to use methods not known in the HSC Syllabus? That would make everything so much easier

Which methods are you talking about? Inspection in general should be fine (might depend on wording of question), and you could differentiate your answer for indefinite integrals if you wanted to show 'working'.

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