HSC 2016 MX2 Integration Marathon (archive) (1 Viewer)

aoc · May 12, 2016

Re: MX2 2016 Integration Marathon

whats reverse quotient rule?

InteGrand · May 12, 2016

Re: MX2 2016 Integration Marathon

aoc said:
whats reverse quotient rule?

It's basically

$$\int \frac{u^\prime v - uv^\prime}{v^2}\text{ d}x = \frac{u}{v}+\text{constant}$ ($u,v$ being functions of $x$).$

aoc · May 12, 2016

Re: MX2 2016 Integration Marathon

is there a reverse product rule

InteGrand · May 12, 2016

Re: MX2 2016 Integration Marathon

aoc said:
is there a reverse product rule

$\noindent Yes: $\int \left(u^\prime v +uv^\prime\right)\text{ d}x =uv + C$.$

leehuan · May 12, 2016

Re: MX2 2016 Integration Marathon

aoc said:
is there a reverse product rule

You can reverse anything that's a differentiation...

Drongoski · May 13, 2016

Re: MX2 2016 Integration Marathon

InteGrand said:
$\noindent Yes: $\int \left(u^\prime v +uv^\prime\right)\text{ d}x =uv + C$.$

Rearranged, this becomes the well-known Integration by Parts:

$\int uv^\prime dx = uv - \int vu^\prime dx$

leehuan · May 17, 2016

Re: MX2 2016 Integration Marathon

Instructive question

$$Find a reduction formula for $I_n=\int_0^x{{\left(1+t^2\right)}^n dt}$

severus snake · May 17, 2016

Re: MX2 2016 Integration Marathon

Help Pls!!!1 in exam rn need help ASAP!!

∫xsecⁿ(x) dx

ps: how to do equations on BOS????/

eyeseeyou · May 17, 2016

Re: MX2 2016 Integration Marathon

severus snake said:
Help Pls!!!1

∫xsecⁿ(x) dx

ps: how to do equations on BOS????/

Use latex

leehuan · May 17, 2016

Re: MX2 2016 Integration Marathon

severus snake said:
Help Pls!!!1 in exam rn need help ASAP!!

∫xsecⁿ(x) dx

ps: how to do equations on BOS????/

You're on your phone during an exam?

severus snake · May 17, 2016

Re: MX2 2016 Integration Marathon

leehuan said:
You're on your phone during an exam?

desktop

SammyT123 · May 19, 2016

Re: MX2 2016 Integration Marathon

severus snake said:
desktop

https://www.youtube.com/watch?v=oTz93Y-qeq0

leehuan · May 23, 2016

Re: MX2 2016 Integration Marathon

Easy question but threw me off

$\int_0^\frac{\pi}{4}{\frac{1-\sin{2x}}{1+\sin{2x}}dx}$

Drsoccerball · May 23, 2016

Re: MX2 2016 Integration Marathon

leehuan said:
Easy question but threw me off

$\int_0^\frac{\pi}{4}{\frac{1-\sin{2x}}{1+\sin{2x}}dx}$

divide by cos^2

Paradoxica · May 23, 2016

Re: MX2 2016 Integration Marathon

Drsoccerball said:
divide by cos^2

It would also be equally fast to long divide and use conjugation of the denominator to obtain sec2x, tan2x and x.

I have no preference for either method though, both are super fast and elegant.

leehuan · May 23, 2016

Re: MX2 2016 Integration Marathon

Funnily enough the original question was just to show that it equalled the integral of tan^2(x) with the same boundaries

InteGrand · May 23, 2016

Re: MX2 2016 Integration Marathon

$\noindent Using the `border-flip' and skipping over some steps, we have$

$\begin{align*}\int _0 ^{\frac{\pi}{4}}\frac{1-\sin 2x}{1+\sin 2x}\text{ d}x &= \int _0 ^{\frac{\pi}{4}}\frac{1-\cos 2x}{1+\cos 2x}\text{ d}x \quad (\text{Pythagorean trig. identity}) \\ &= \int _0 ^{\frac{\pi}{4}}\tan^2 x\text{ d}x \quad (\text{identity from that other recent thread (a proof linked below))}) \\ &= 1 - \frac{\pi}{4}\quad (\text{inspection (use }\tan^2 x = \sec^2 x - 1\text{ to find a primitive})). \end{align*}$

(Identity used: http://community.boredofstudies.org...do-we-need-memorise-identity.html#post7159341)

leehuan · May 23, 2016

Re: MX2 2016 Integration Marathon

Basically yep^

kawaiipotato · May 26, 2016

Re: MX2 2016 Integration Marathon

$\int \frac{\sqrt{x}}{\sqrt{a^3 - x^3 }} dx$

$where a is a constant$

Paradoxica · May 27, 2016

MX2 2016 Integration Marathon

kawaiipotato said:
$\int \frac{\sqrt{x}}{\sqrt{a^3 - x^3 }} dx$

$where a is a constant$

Recollecting the differential on the numerator yields

$\frac{2}{3} \int \frac{\text{d}(x\sqrt{x})}{\sqrt{a^3-x^3}}$

Which is easy to see as a standard inverse trigonometric function

$\frac{2}{3}\sin^{-1}{\sqrt\frac{x^3}{a^3}}$

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