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

Shadowless · Aug 8, 2012

Re: MX2 Integration Marathon

How do u get those integral symbols

rolpsy · Aug 8, 2012

Re: MX2 Integration Marathon

nightweaver066 said:
$\int^{\frac{\pi}{2}}_0 \frac{x}{tanx}dx$

$= \int^{\frac{\pi}{2}}_0 xcotxdx$

$= [xln|sinx|]^{\frac{\pi}{2}}_0 - \int^{\frac{\pi}{2}}_0 ln|sinx|dx$

Now IIRC, that second integral is easily done by 2 applications of IBP and rearrangements.

D: i just realised its undefined at x = 0.. L'Hopital's rule?

yup

SpiralFlex · Aug 8, 2012

Re: MX2 Integration Marathon

SpiralFlex said:
$$Show that if$\; \int \frac{1}{uf(u)}\; du=F(u)+C, \;$then$\; \int \frac{m}{xf(x^m)}}\;dx = F(x^m)+C$

$Find:$

$$i)$\; \int \frac{1}{x^n-x}\;dx$

$$ii)$\; \int \frac{1}{\sqrt{x^n+x^2}}\;dx$

.

asianese · Aug 8, 2012

Re: MX2 Integration Marathon

Yeah I ran into trouble with the limits...LHopitals it is!

U MAD BRO · Aug 9, 2012

Re: MX2 Integration Marathon

rolpsy said:
lol you forgot to integrate!

$\int^{\frac{\pi}{4}}_{0} \ln(2) \, \mathrm{d}x = \frac{\pi}{4}\ln(2)$

which gives the correct answer

here is a really difficult one:

$\int^{\frac{\pi}{2}}_{0} \frac{x}{\tan(x)} \, \mathrm{d}x$

good luck

lol forgot to integrate, thank u, i fixed it

ill have a go at ur soon! once i finish studying for mx1.

U MAD BRO · Aug 9, 2012

Re: MX2 Integration Marathon

How do you evaluate an integral "numerically" ???

U MAD BRO · Aug 9, 2012

Re: MX2 Integration Marathon

$\int_{0}^{\infty }\frac{cos(x)}{1+(x)^{2}}dx$

good luck!

U MAD BRO · Aug 9, 2012

Re: MX2 Integration Marathon

$\int_{0}^{\infty }e^{-(x)^{2}}dx$

U MAD BRO · Aug 9, 2012

Re: MX2 Integration Marathon

$\int_{0}^{1}loglog\left ( \frac{1}{x} \right )dx$

good luck!

U MAD BRO · Aug 9, 2012

Re: MX2 Integration Marathon

$\int_{0}^{\infty }\frac{x}{e^{x}-1}dx$

U MAD BRO · Aug 9, 2012

Re: MX2 Integration Marathon

$\int_{-\infty }^{\infty }\frac{sin(x)}{x}dx$

seanieg89 · Aug 9, 2012

Re: MX2 Integration Marathon

U MAD BRO keep your integrals at the level of the MX2 integration topic. If you are interested in how to do these yourself, do outside research / ask in extracurricular. This thread is an inappropriate place to post them.

@Rolpsy nice question!

$I=\int_0^{\pi/2}x\cot(x)\,dx=\int_0^{\pi/2}(\pi/2-x)\tan(x)\,dx\\ \Rightarrow 2I=\int_0^{\pi/2}\frac{2x\cos^2(x)+2(\pi/2-x)\sin^2(x)}{2\sin(x)\cos(x)}\, dx\\=\int_0^{\pi/2}\frac{2x\cos(2x)+\frac{\pi}{2}(1-\cos(2x))}{\sin(2x)}\,dx\\\Rightarrow 4I=\int_0^{\pi}\frac{x\cos(x)+\frac{\pi}{2}(1-\cos(x))}{\sin(x)}\, dx\\=\int_0^{\pi/2}x\cot(x)\,dx+\frac{\pi}{2}\int_0^{\pi/2}\csc(x)-\cot(x)\,dx+\int_{\pi/2}^\pi (x-\pi)\cot(x)\,dx+\frac{\pi}{2}\int_{\pi/2}^\pi\csc(x)+\cot(x)\,dx\\=2I+\pi\int_0^{\pi/2}\csc(x)-\cot(x)\,dx\\=2I-\pi\log|1+\cos(x)|]^{\pi/2}_0\\\Rightarrow I=\frac{\pi}{2}\log(2).$

deswa1 · Aug 9, 2012

Re: MX2 Integration Marathon

Fuuark this thread looks so interesting. Will get involved soon

Johnstan · Aug 9, 2012

Re: MX2 Integration Marathon

deswa1 said:
Fuuark this thread looks so interesting. Will get involved soon

best be getting involved brah

U MAD BRO · Aug 9, 2012

Re: MX2 Integration Marathon

=(
everything is MX2 level, in fact, MX2 syllabub is unlimited

can u show us how to do these bad ass integrals Sean, please

Carrotsticks · Aug 9, 2012

Re: MX2 Integration Marathon

You will never be asked straight out to evaluate special integrals like the Logarithmic, Sine, Gaussian Integrals all in one go. If anything, they will lead you into it.

asianese · Aug 9, 2012

Re: MX2 Integration Marathon

Yeah, no, UMAD/TheMatrix, no.

U MAD BRO · Aug 9, 2012

Re: MX2 Integration Marathon

asianese said:
Yeah, no, UMAD/TheMatrix, no.

BTW everyone knows that... so what?

Carrotsticks · Aug 10, 2012

Re: MX2 Integration Marathon

U MAD BRO said:
How do you evaluate an integral "numerically" ???

Sometimes, numerical values are assigned to integrals that would otherwise have no value because the domain of integration contains a point of singularity or a discontinuity.

In which case, a Cauchy Principle Value is assigned.

But like seanieg89 said, try to keep your integrals relatively elementary in nature.

U MAD BRO · Aug 10, 2012

Re: MX2 Integration Marathon

Carrotsticks said:
Sometimes, numerical values are assigned to integrals that would otherwise have no value because the domain of integration contains a point of singularity or a discontinuity.

In which case, a Cauchy Principle Value is assigned.

But like seanieg89 said, try to keep your integrals relatively elementary in nature.

I'm really interested to know how to do these integrals, I'll post them in extracurricular =)

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

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