Higher Level Integration Marathon & Questions (2 Viewers)

InteGrand · Jun 29, 2016

Re: Extracurricular Integration Marathon

So is this integral from an IB maths textbook?

Paradoxica · Jun 29, 2016

Re: Extracurricular Integration Marathon

InteGrand said:
So is this integral from an IB maths textbook?

According to Leehuan

InteGrand · Jun 29, 2016

Re: Extracurricular Integration Marathon

Paradoxica said:
According to Leehuan

What techniques of integration do they learn in IB? Do they learn Complex Analysis methods? I'm guessing they'd do double integrals?

leehuan · Jun 29, 2016

Re: Extracurricular Integration Marathon

Still waiting for my source to reply...

Paradoxica · Jun 29, 2016

Re: Extracurricular Integration Marathon

seanieg89 said:
We are in the extracurricular marathon, so post a contour integration solution if you do find one.

The slow decay of the integrand again makes things nontrivial (what contour do you suggest?), and the singularity is removable, not a pole so there is no benefit in contouring about it (for the original function, it is a pole of the complexified function, but the problem of slow decay remains).

The classical semicircular contour indented at the origin works for every power series term (Jordan's Lemma) but the problem still remains at bay: How do we justify interchange of summation and integration?

leehuan · Jun 30, 2016

Re: Extracurricular Integration Marathon

Haven't gotten a proper answer but right now he said some challenge question from "Quizlet"

InteGrand · Jun 30, 2016

Re: Extracurricular Integration Marathon

leehuan said:
Haven't gotten a proper answer but right now he said some challenge question from "Quizlet"

Is Quizlet an online thing, or paper-based?

leehuan · Jun 30, 2016

Re: Extracurricular Integration Marathon

InteGrand said:
Is Quizlet an online thing, or paper-based?

Turns out it was an...app

InteGrand · Jun 30, 2016

Re: Extracurricular Integration Marathon

leehuan said:
Turns out it was an...app

Nice. So it's not actually IB-related?

leehuan · Jun 30, 2016

Re: Extracurricular Integration Marathon

InteGrand said:
Nice. So it's not actually IB-related?

My assumption is that it was tagged as an IB challenge question. (That's what he implied at least)

Paradoxica · Jul 3, 2016

Re: Extracurricular Integration Marathon

Without invoking the Dilogarithm, or using infinite sums of power series, evaluate

$\int_0^1 \frac{\log{(x^2+1)}}{x+1}$d$x$

Mongoose528 · Jul 3, 2016

Re: Extracurricular Integration Marathon

Is this a year 12 topic?

Paradoxica · Jul 3, 2016

Re: Extracurricular Integration Marathon

Mongose528 said:
Is this a year 12 topic?

No, the Dilogarithm is beyond the curriculum, as is practically everything else on this thread.

leehuan · Jul 3, 2016

Re: Extracurricular Integration Marathon

Mongose528 said:
Is this a year 12 topic?

Integration is a topic in Year 12, however the extent to which it goes is not massive. Even the MX2 Integration Marathon tends to feature questions that are beyond what's examinable in the HSC, let alone the stuff here.

Paradoxica · Jul 3, 2016

Re: Extracurricular Integration Marathon

Here's an easier one. Fun Exercise

$\int_0^1 \frac{\tan^{-1}{x}\text{d}x}{x\sqrt{1-x^2}}$

Paradoxica · Jul 4, 2016

Extracurricular Integration Marathon

seanieg89 said:
There has to be a less potentially dodgy way of doing it, I also thought of expanding out a series in e^(ix) to start with, but I couldn't see any way of justifying that so I left it.

Gotta be careful with these things, because functions/sequences that oscillate and decay but are not absolutely integrable are a common source of counterexamples to otherwise believable claims.

Fourier expansion. Is that less dodgy, or more?

Also something about the Zeta Regularisation technique.

seanieg89 · Jul 4, 2016

Re: Extracurricular Integration Marathon

Paradoxica said:
Fourier expansion. Is that less dodgy, or more?

Also something about the Zeta Regularisation technique.

Well a power series in e^(ix) IS a Fourier series. The difficulty is in getting a mode of convergence that allows us to commute the limit with the conditionally convergent integral. I did think of an idea that probably deals with it though, and will post it during an afternoon break today if it works.

Paradoxica · Jul 4, 2016

Re: Extracurricular Integration Marathon

Find this limit:

$\lim_{x \to 0^+} (e^{-x^2}-e^{-x})\ln x$

Hence, evaluate this:

$\int_0^{\infty} \frac{e^{-x^2}-e^x}{x}$d$x$

Paradoxica · Jul 4, 2016

Re: Extracurricular Integration Marathon

A fairly easy exercise.

$\int_{-\infty}^{\infty} \frac{\cos{ax} - \cos{bx}}{x^2} $d$x$, where ${a,b} \in \mathbb{R}^+$

Paradoxica · Jul 4, 2016

Re: Extracurricular Integration Marathon

seanieg89 said:
Well a power series in e^(ix) IS a Fourier series. The difficulty is in getting a mode of convergence that allows us to commute the limit with the conditionally convergent integral. I did think of an idea that probably deals with it though, and will post it during an afternoon break today if it works.

Well the thing I was looking at for Zeta Regularisation was to use the method I used back in the early days of this thread

http://community.boredofstudies.org...ricular-integration-marathon.html#post7092858

But the thing is the justification...

Higher Level Integration Marathon & Questions (2 Viewers)

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