HSC 2012-14 MX2 Integration Marathon (archive) (1 Viewer)

seanieg89 · Mar 9, 2014

Re: MX2 Integration Marathon

In this question we compute the Dirichlet integral using high school methods.

$1. Show that: \\ \\$\int_0^{\pi/2} \frac{\sin((2n+1)x)}{\sin(x)}\, dx = \frac{\pi}{2}.$\\ \\ 2. Let: \\ \\$I_n:=\int_0^{\frac{(2n+1)\pi}{2}}\frac{\sin(x)}{x}\, dx$\\ and show that $D_n=\frac{\pi}{2}-I_n \rightarrow 0$ by using 1.\\ \\ 3. Hence, given that the limit: $I:=\lim_{T\rightarrow \infty}\left(\int_0^T \frac{\sin(x)}{x}\, dx\right) $ exists, find it.$

Note: n ranges over the non-negative integers.

Sy123 · Mar 9, 2014

Re: MX2 Integration Marathon

seanieg89 said:
$\int \sqrt[3]{\tan(x)}\, dx$

$\tan x = u^3 \Rightarrow 3\int \frac{u^2\cdot u}{1+u^6} \ du$

$z=u^2 \Rightarrow \frac{3}{2}\int \frac{z}{1+z^3} \ dz$

$(z^3+1) = (z+1)(z^2-z+1) \ \ $etc etc$$

seanieg89 said:
In this question we compute the Dirichlet integral using high school methods.

$1. Show that: \\ \\$\int_0^{\pi/2} \frac{\sin((2n+1)x)}{\sin(x)}\, dx = \frac{\pi}{2}.$\\ \\ 2. Let: \\ \\$I_n:=\int_0^{\frac{(2n+1)\pi}{2}}\frac{\sin(x)}{x}\, dx$\\ and show that $D_n=\frac{\pi}{2}-I_n \rightarrow 0$ by using 1.\\ \\ 3. Hence, given that the limit: $I:=\lim_{T\rightarrow \infty}\left(\int_0^T \frac{\sin(x)}{x}\, dx\right) $ exists, find it.$

Note: n ranges over the non-negative integers.

Dayum son

Sy123 · Mar 9, 2014

Re: MX2 Integration Marathon

$\int \frac{1}{1-e^{kx}} \ dx$

Sy123 · Mar 9, 2014

Re: MX2 Integration Marathon

$\int \sqrt{\csc x - \sin x} \ dx$

seanieg89 · Mar 9, 2014

Re: MX2 Integration Marathon

Sy123 said:
$\tan x = u^3 \Rightarrow 3\int \frac{u^2\cdot u}{1+u^6} \ du$

$z=u^2 \Rightarrow \frac{3}{2}\int \frac{z}{1+z^3} \ dz$

$(z^3+1) = (z+1)(z^2-z+1) \ \ $etc etc$$

Dayum son

Yep. One of those ones where the difficulty is more mechanical than conceptual. (Worthwhile for any 2014'ers to crunch through the working and compare their answers with wolfram alpha's).

Dayum as in a cool proof or a hard question? I wasn't sure how much leading to do without giving it away.

Sy123 · Mar 9, 2014

Re: MX2 Integration Marathon

$\int \frac{1}{x^n - x} \ dx \ \ \ (n>1)$

Sy123 · Mar 9, 2014

Re: MX2 Integration Marathon

$\int e^{e^x+x} \ dx$

Sy123 · Mar 9, 2014

Re: MX2 Integration Marathon

seanieg89 said:
Dayum as in a cool proof or a hard question? I wasn't sure how much leading to do without giving it away.

Well I mean't that it was cool, not sure if its difficult haven't tried it yet (probably is difficult considering the other questions you post lol)

seanieg89 · Mar 9, 2014

Re: MX2 Integration Marathon

Sy123 said:
Well I mean't that it was cool, not sure if its difficult haven't tried it yet (probably is difficult considering the other questions you post lol)

Lol. It's on the easier side of the ones that I post. There is just one thing that might be slightly difficult to spot (*), but the rest is okay.

Edit: I also take no credit for this proof, just happened to stumble across it the other day whilst reading things on the internet. It's strikingly elementary compared to most evaluations of that integral.

(*) Well, two things actually.

1. I won't spoil, it is something an MX2 student is very familiar with but they may not know it is useful in this particular situation.

2. I am guessing most students will overlook this, but looking at the limit of (1/sin(x)-1/x) as x-> 0+ is not entirely straightforward. It is enough to use x-sin(x) =< x^3/6 for positive x, which can be proven by repeated integration of (1-cos(x)) >= 0.

hit patel · Mar 10, 2014

Re: MX2 Integration Marathon

Hey does anyone have the compiled copy of all these questions. Rumour was spread that int. marathon question copies were being done?

Carrotsticks · Mar 11, 2014

Re: MX2 Integration Marathon

HeroicPandas said:
Integral of e^[e^(x) +x].dx

Sy123 said:
$\int e^{e^x+x} \ dx$

Haha and HeroicPandas liked it too.

Sy123 · Mar 11, 2014

Re: MX2 Integration Marathon

Carrotsticks said:
Haha and HeroicPandas liked it too.

I think this integral has been posted at least 5 times lol

Sy123 · Mar 17, 2014

Re: MX2 Integration Marathon

Sy123 said:
$\int \frac{1}{1-e^{kx}} \ dx$

$\int \frac{dx}{1-e^{kx}} \ dx = \int \frac{e^{-kx}}{e^{-kx}-1} \ dx = \frac{-1}{k}\ln |e^{-kx}-1| + c$

-------------------------------------------------------

$\\ $Find$ \ \ I_n = \int_0^{\pi} e^{x} \sin^{2n}x \ dx$

mathsbrain · Mar 17, 2014

Re: MX2 Integration Marathon

Sy123 said:
$\int \frac{dx}{1-e^{kx}} \ dx = \int \frac{e^{-kx}}{e^{-kx}-1} \ dx = \frac{-1}{k}\ln |e^{-kx}-1| + c$

-------------------------------------------------------

$\\ $Find$ \ \ I_n = \int_0^{\pi} e^{x} \sin^{2n}x \ dx$

Ok, it's by parts twice, letting primitive be e^x in both times.too much tex to type...

Sy123 · Mar 21, 2014

Re: MX2 Integration Marathon

$\int_{-\infty}^{\infty} \frac{dx}{(x^2+a^2)(x^2+b^2)}$

braintic · Mar 21, 2014

Re: MX2 Integration Marathon

Sy123 said:
$\int_{-\infty}^{\infty} \frac{dx}{(x^2+a^2)(x^2+b^2)}$

Just wondering ... in uni maths, do they write infinities in the limits like this, or do they write these integrals using limit notation (different usage of the word 'limit')?

Carrotsticks · Mar 21, 2014

Re: MX2 Integration Marathon

braintic said:
Just wondering ... in uni maths, do they write infinities in the limits like this, or do they write these integrals using limit notation (different usage of the word 'limit')?

We write infinities in the limits, very popular in Contour Integration.

http://en.wikipedia.org/wiki/Methods_of_contour_integration

The only time you would make the limits say 0 to N, and then make N -> infinity, is if you're analysing the limit as a value that the sequence (the integral from 0 to N) approaches. So this approach would be used say if you're finding the limiting area under the curve y=1/x^2 from 1 onwards.

But there are methods of integration where you evaluate from - infinity to infinity directly, so the whole limit to N thing is unnecessary.

seanieg89 · Mar 21, 2014

Re: MX2 Integration Marathon

The meaning of infinities in the limits of the Riemann integral (ie improper integrals) is defined by the the limit notation you speak of braintic.

$\int_a^\infty f(x)\, dx := \lim_{T\rightarrow\infty}\left(\int_a^T f(x)\, dx\right).$

Shadowdude · Mar 21, 2014

Re: MX2 Integration Marathon

Sy123 said:
$\int_{-\infty}^{\infty} \frac{dx}{(x^2+a^2)(x^2+b^2)}$

Are improper integrals in the MX2 syllabus now?!

Sy123 · Mar 21, 2014

Re: MX2 Integration Marathon

Shadowdude said:
Are improper integrals in the MX2 syllabus now?!

There you go mate

$\lim_{N \to \infty} \int_{-N}^N \frac{dx}{(x^2+a^2)(x^2+b^2)}$

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