HSC 2012-14 MX2 Integration Marathon (archive) (4 Viewers)

dunjaaa · Apr 2, 2014

Re: MX2 Integration Marathon

Just sat my half yearly for MX2 today, this was my recurrence question. I reckon it was a good one

Screen shot 2014-04-02 at 7.32.13 PM.png

Sy123 · Apr 2, 2014

Re: MX2 Integration Marathon

Let's generalize that just a little bit

$\\ $Find the re-currence formula in$ \ n \ $for$ \\ \\ I_n = \int_0^1 x^n (1-x^m)^{1/k} \ dx \ \ $for$ \ n > (m+1) \ $for naturals$ \ n$

(its doable I tried it)

Ikki · Apr 3, 2014

Re: MX2 Integration Marathon

Sy123 said:
Let's generalize that just a little bit

$\\ $Find the re-currence formula in$ \ n \ $for$ \\ \\ I_n = \int_0^1 x^n (1-x^m)^{1/k} \ dx \ \ $for$ \ n > (m+1) \ $for naturals$ \ n$

(its doable I tried it)

Ofcourse you did.

dunjaaa · Apr 3, 2014

Re: MX2 Integration Marathon

chucking a carrot I see
I got I(n)={[k(n-m+1)]/[m+k(n+1)]}I(n-m)

Sy123 · Apr 4, 2014

Re: MX2 Integration Marathon

dunjaaa said:
chucking a carrot I see
I got I(n)={[k(n-m+1)]/[m+k(n+1)]}I(n-m)

Yep something like that

integral95 · Apr 4, 2014

Re: MX2 Integration Marathon

Hey guys in dat UNSW ASOC integration bee, there was this weird question (carrot would approve)

$\int \sqrt{x+\sqrt{x+\sqrt{x+...}}} dx$

Carrotsticks · Apr 4, 2014

Re: MX2 Integration Marathon

MethewYan said:
Hey guys in dat UNSW ASOC integration bee, there was this weird question (carrot would approve)

$\int \sqrt{x+\sqrt{x+\sqrt{x+...}}} dx$

I thought it was a rather different one though lol.

The actual integration is very straightforward, it's just finding the closed form that not too many people are familiar with.

Sy123 · Apr 4, 2014

Re: MX2 Integration Marathon

$\int_0^{\pi/2}\frac{x}{\sin x + \cos x} \ dx$

dunjaaa · Apr 4, 2014

Re: MX2 Integration Marathon

Screen shot 2014-04-04 at 11.44.02 PM.png

Sy123 · Apr 5, 2014

Re: MX2 Integration Marathon

MethewYan said:
Hey guys in dat UNSW ASOC integration bee, there was this weird question (carrot would approve)

$\int \sqrt{x+\sqrt{x+\sqrt{x+...}}} dx$

Hint: Let u = (that expression)

dunjaaa · Apr 5, 2014

Re: MX2 Integration Marathon

Screen shot 2014-04-05 at 4.35.33 PM.png

Not sure if correct

Sy123 · Apr 5, 2014

Re: MX2 Integration Marathon

dunjaaa said:
View attachment 30228 Not sure if correct

Yep well done!

Carrotsticks · Apr 5, 2014

Re: MX2 Integration Marathon

dunjaaa said:
View attachment 30228 Not sure if correct

You could also do this.

$\\ u=\sqrt{x+\sqrt{x+\sqrt{x+...}}} \\\\ u^2 = x+u \\\\ u^2 - u - x = 0 \\\\ u=\frac{1 \pm \sqrt{1+4x}}{2} $ and then just integrate this instead.$$

Sy123 · Apr 5, 2014

Re: MX2 Integration Marathon

$\int_0^{\pi/4} \frac{x + \sin x}{1 + \cos x} \ dx$

dunjaaa · Apr 5, 2014

Re: MX2 Integration Marathon

change into half angles -> int{[x+2sin(x/2)cos(x/2)]/2cos^2(x/2) dx}
=int{1/2 [xsec^2(x/2)]+tan(x/2) dx}
=[xtan(x/2)] between x=0 and x=pi/4 by reverse product rule
=pi/4tan(pi/8)

Sy123 · Apr 5, 2014

Re: MX2 Integration Marathon

dunjaaa said:
change into half angles -> int{[x+2sin(x/2)cos(x/2)]/2cos^2(x/2) dx}
=int{1/2 [xsec^2(x/2)]+tan(x/2) dx}
=[xtan(x/2)] between x=0 and x=pi/4 by reverse product rule
=pi/4tan(pi/8)

nice work

$\int_0^{\pi/6} \frac{\sqrt{1 + \sin x}}{\cos x} \ dx$

Sy123 · Apr 5, 2014

Re: MX2 Integration Marathon

$\int_0^{\pi} \frac{\sin^3x}{\sin^2x + 8} \ dx$

Davo_01 · Apr 9, 2014

Re: MX2 Integration Marathon

Sy123 said:
nice work

$\int_0^{\pi/6} \frac{\sqrt{1 + \sin x}}{\cos x} \ dx$

Sy123 · Apr 10, 2014

Re: MX2 Integration Marathon

$\\ $For the positive real$ \ x \ $find the minimum value of$ \ \ f(x) = \int_x^{2x} t\ln t - t \ dt$

hit patel · Apr 10, 2014

Re: MX2 Integration Marathon

Sy123 said:
$\\ $For the positive real$ \ x \ $find the minimum value of$ \ \ f(x) = \int_x^{2x} t\ln t - t \ dt$

e^(4/9) ? Looks way off.

HSC 2012-14 MX2 Integration Marathon (archive) (4 Viewers)

Active Member

Attachments

This too shall pass

Active Member

Active Member

This too shall pass

Well-Known Member

Retired

This too shall pass

Active Member

This too shall pass

Active Member

This too shall pass

Retired

This too shall pass

Active Member

This too shall pass

This too shall pass

Active Member

This too shall pass

New Member

Users Who Are Viewing This Thread (Users: 0, Guests: 4)