HSC 2015 MX2 Integration Marathon (archive) (1 Viewer)

VBN2470 · Apr 4, 2015

Re: MX2 2015 Integration Marathon

nightweaver066 said:
I see you're on an integration bee spree with your questions haha

Drsoccerball · Apr 4, 2015

Re: MX2 2015 Integration Marathon

New question

$\int{\pi^{x}}dx$

VBN2470 · Apr 4, 2015

Re: MX2 2015 Integration Marathon

Drsoccerball said:
New question
$\int{\pi^{x}}dx$

$NOTE that $\pi^{x} = e^{(\ln{\pi})x}$, then use standard laws of integration of exponentials to get \\ $\frac{\pi^{x}}{\ln\pi}+C$$

VBN2470 · Apr 4, 2015

Re: MX2 2015 Integration Marathon

VBN2470 said:
NEW QUESTION:

$Evaluate $\int \frac{\sec\theta+\tan\theta}{\csc\theta+\cot\theta}d\theta$$

BUMP

Drsoccerball · Apr 4, 2015

Re: MX2 2015 Integration Marathon

VBN2470 said:
BUMP

i answered this

VBN2470 · Apr 4, 2015

Re: MX2 2015 Integration Marathon

NEW QUESTION:

$Evaluate $\int_{0}^{\infty} \frac{t^2}{(1+t)^5}dt$$

Drsoccerball · Apr 4, 2015

Re: MX2 2015 Integration Marathon

VBN2470 said:
NEW QUESTION:

$Evaluate $\int_{0}^{\infty} \frac{t^2}{(1+t)^5}dt$$

$let u=1 +t$

$du=dt$

$\int_1^{inf}{ \frac{u^2-2u +1}{u^5}}$

$[\frac{-1}{2u^2} - \frac{2}{3u^3} +ln(u)]$

$ln(infinity) -\frac{1}{6}$

This seems wrong ahaha

VBN2470 · Apr 4, 2015

Re: MX2 2015 Integration Marathon

Drsoccerball said:
$let u=1 +t$

$du=dt$

$\int_1^{inf}{ \frac{u^2-2u +1}{u^5}}$

$[\frac{-1}{2u^2} - \frac{2}{3u^3} +ln(u)]$

$ln(infinity) -\frac{1}{6}$

This seems wrong ahaha

Almost there, I think you integrated the last part incorrectly, where the ln(u) comes from? You may have also mixed up some of the signs in your expression as well..

Drsoccerball · Apr 4, 2015

Re: MX2 2015 Integration Marathon

let u=1 +t [/tex]

$du=dt$

$\int_1^{inf}{ \frac{u^2-2u +1}{u^5}}$

$\int{\frac{1}{u^3} - \frac{2}{u^4} + \frac{1}{u^5} }$

$\frac{-1}{2u^2}+\frac{2}{3u^3} - \frac{1}{4u^4}$

$\frac{1}{12}$

Drsoccerball · Apr 4, 2015

Re: MX2 2015 Integration Marathon

$\int{e^{x^2}}$

VBN2470 · Apr 4, 2015

Re: MX2 2015 Integration Marathon

Drsoccerball said:
$\int{e^{x^2}}$

Don't know if this can be done within MX2 bounds..

Drsoccerball · Apr 4, 2015

Re: MX2 2015 Integration Marathon

VBN2470 said:
Don't know if this can be done within MX2 bounds..

Whats the uni method ?

Carrotsticks · Apr 4, 2015

Re: MX2 2015 Integration Marathon

Drsoccerball said:
$\int{e^{x^2}}$

No elementary solution.

Keep in mind (if you're making up these integrals) that many expressions don't have integrals that can be expressed in elementary terms.

InteGrand · Apr 4, 2015

Re: MX2 2015 Integration Marathon

Drsoccerball said:
New question:
$\int_0^1{\frac{x^7-1}{logx}}$

$\int_{0}^{1}\frac{x^7 -1}{\ln x}\text{ d}x=\int_{0}^{1} \left(\int_{0}^{7}x^t\text{ d}t \right )\text{ d}x$

$=\int_{0}^{7} \left( \int_{0}^{1}x^t\text{ d}x\right ) \text{ d}t$ , from Fubini's theorem

$=\int_{0}^{7} \left [ \frac{1}{t+1}x^{t+1} \right ]_{x=0} ^{x=1} \text{ d}t$

$=\int_{0}^{7} \frac{1}{t+1} \text{ d}t$

$=\left [ \ln (t+1) \right ]_0 ^7$

$=\ln 8 - \ln 1$

$=2\ln 3$ .

I can't see a way to do this using HSC 4U methods.

VBN2470 · Apr 4, 2015

Re: MX2 2015 Integration Marathon

InteGrand said:
$\int_{0}^{1}\frac{x^7 -1}{\ln x}\text{ d}x=\int_{0}^{1} \left(\int_{0}^{7}x^t\text{ d}t \right )\text{ d}x$

$=\int_{0}^{7} \left( \int_{0}^{1}x^t\text{ d}x\right ) \text{ d}t$ , from Fubini's theorem

$=\int_{0}^{7} \left [ \frac{1}{t+1}x^{t+1} \right ]_{x=0} ^{x=1} \text{ d}t$

$=\int_{0}^{7} \frac{1}{t+1} \text{ d}t$

$=\left [ \ln (t+1) \right ]_0 ^7$

$=\ln 8 - \ln 1$

$=2\ln 3$ .

I can't see a way to do this using HSC 4U methods.

Damn that's neat!

Drsoccerball · Apr 4, 2015

Re: MX2 2015 Integration Marathon

Carrotsticks said:
No elementary solution.

Keep in mind (if you're making up these integrals) that many expressions don't have integrals that can be expressed in elementary terms.

I havnt made up an integral on this forum i found these in textbooks, test papers, ect..

Carrotsticks · Apr 4, 2015

Re: MX2 2015 Integration Marathon

Drsoccerball said:
I havnt made up an integral on this forum i found these in textbooks, test papers, ect..

So where did you find it then?

Drsoccerball · Apr 4, 2015

Re: MX2 2015 Integration Marathon

Carrotsticks said:
So where did you find it then?

it was an integration bee

Carrotsticks · Apr 4, 2015

Re: MX2 2015 Integration Marathon

Drsoccerball said:
it was an integration bee

Err okay, were students expected to produce a non-elementary result? Kinda defeats the purpose of an integration bee.

Drsoccerball · Apr 4, 2015

Re: MX2 2015 Integration Marathon

Carrotsticks said:
Err okay, were students expected to produce a non-elementary result? Kinda defeats the purpose of an integration bee.

Are they directed at people at uni

?

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