Difficult Maths/Physics Problems Help Thread

I've got what I think is a fairly tricky question, though I didn't actually get to try it as the solution was right next to the integral, so that kind of spoiled it for me.

Show that if

f

is any continuous function and

n>0

,

\displaystyle \int^n_{n^{-1}} f(x+\dfrac{1}{x})\dfrac{log x}{x} \ dx = 0

Reply 202

Necrosyrtes

4

Original post by Gawain

Somewhat related to the topic of summations and integrals.

\int_{0}^{\infty}\left \lfloor x \right \rfloor e^{-x}dx

Spoiler

Reply 203

jjpneed1

2

Original post by ThatPerson

I've got what I think is a fairly tricky question, though I didn't actually get to try it as the solution was right next to the integral, so that kind of spoiled it for me.

Show that if

f

is any continuous function and

n>0

,

\displaystyle \int^n_{n^{-1}} f(x+\dfrac{1}{x})\dfrac{log x}{x} \ dx = 0

Spoiler

Reply 204

ThatPerson

Original post by jjpneed1

Spoiler

Reply 205

jjpneed1

2

Original post by ThatPerson

Spoiler

Reply 206

ThatPerson

Original post by jjpneed1

Spoiler

Ah, that's quite useful. Can you spot to use that sub by noting that the limits reverse?

Reply 207

jjpneed1

2

Original post by ThatPerson

Ah, that's quite useful. Can you spot to use that sub by noting that the limits reverse?

Yeah that's generally when it's used. Alternatively if you have an integral from 0 to infinity then you can rewrite it as an integral from 0 to 1 added to an integral from 1 to infinity. Then you can use the sub on the second integral to transform it into an integral from 0 to 1 and then recombine it with the first integral to get an overall integral from 0 to 1 which is hopefully simpler than the original.

Reply 208

Just did this one.

Prove for any parallelepiped that all diagonals intersect at a common point.

Reply 209

Original post by Blazar

Just did this one.

Prove for any parallelepiped that all diagonals intersect at a common point.

what is a parallepiped??

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Reply 210

Original post by physicsmaths

what is a parallepiped??

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Google is your friend.

Reply 211

Original post by Blazar

Google is your friend.

lol ok

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Reply 212

Kvothe the Arcane

20

Original post by physicsmaths

lol ok

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3d shape formed from parallelograms. The more commonly known cubes and cuboids are special parallelepipeds.

(edited 9 years ago)

Reply 213

Original post by physicsmaths

lol ok

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Sorry, I was really busy there.

Reply 214

Original post by Blazar

Just did this one.

Prove for any parallelepiped that all diagonals intersect at a common point.

Hmm I've used vectors. Is this the right approach.

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Reply 215

Original post by physicsmaths

Hmm I've used vectors. Is this the right approach.

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That's what I did. There are probably other ways to prove it but I used the vector approach...

Reply 216

Krollo

17

Original post by physicsmaths

Hmm I've used vectors. Is this the right approach.

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I haven't done this but I imagine it is easiest to turn it into a vectors / coordinate geometry bash. There may well be an elegant geometric argument but I fail at visualising 3D stuff.

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Reply 217