The Proof is Trivial!

if youre referring to the original integral, i think he also changed the signs as well to change the limits, just for ease of manipulation id imagine

Changing order of limits is often not justified though.

Reply 2781

newblood

19

Original post by james22

Changing order of limits is often not justified though.

i havent done uni maths so i'll step away here :tongue:

I know this isnt a maths help thread and i dont wish to derail it but since people are currently online and no-one has posted solutions recently : can anyone provide some clarity on this thread;

http://www.thestudentroom.co.uk/showthread.php?t=2600276

i can not understand why its not applying to differentiation/calculus - is that in fact true?

Reply 2782

Lord of the Flies

19

Original post by james22

Why are you justified in swapping the order of limits here?

DCT of course.

(

use

\delta_{n,k}=a_{n,2k-1}-a_{n,2k}<k^{-2}

where

a_{n,m}=n\big(mn-1\big)^{-1}

for instance

)

(edited 10 years ago)

Reply 2783

cheeseman552

If I drop a cannon ball off the Eiffel Tower, where does it land? :wink:

Reply 2784

james22

16

Original post by cheeseman552

If I drop a cannon ball off the Eiffel Tower, where does it land? :wink:

On the ground.

Reply 2785

Smaug123

15

Original post by cheeseman552

If I drop a cannon ball off the Eiffel Tower, where does it land? :wink:

By the way, a really really neat proof that speed of falling is independent of mass is:
Drop two identical cannonballs next to each other. They will obviously fall at the same speed as each other.
Repeat the experiment but this time tie the cannonballs together with a loose piece of string.

Reply 2786

interstitial

18

Original post by cheeseman552

If I drop a cannon ball off the Eiffel Tower, where does it land? :wink:

It doesn't... O_o

Posted from TSR Mobile

Reply 2787

james22

16

In how many different ways can you find

\displaystyle\int e^x \cos(x) \cosh(x) dx

Try doing it using only the cyclic nature of teh derivatives of the functions in the integrand.

(edited 10 years ago)

Reply 2788

Tarquin Digby

9

Original post by james22

In how many different ways can you find

\displaystyle\int e^x \cos(x) \cosh(x) dx

Try doing it using only the cyclic nature of teh derivatives of the functions in the integrand.

Unparseable latex formula:

[br]\displaystyle\quad \int e^x\cos{x}\cosh{x}\,\mathrm{d}x\\[br]\displaystyle=\Re\:\frac{1}{2} \int e^{(2+i)x}+e^{ix}\,\mathrm{d}x\\[br]\displaystyle=\Re\:\frac{e^{(2+i)x}}{2(2+i)}+\frac{e^{ix}}{2i}+C\\[br]\displaystyle=\Re\:\frac{(2-i)e^{(2+i)x}}{10}-\frac{ie^{ix}}{2}+C\\[br]\displaystyle=\frac{\sin x}{2}+\frac{e^{2x}\cos{x}}{5}+ \frac{e^{2x}\sin{x}}{10}+C[br]

Reply 2789

Smaug123

15

Original post by james22

In how many different ways can you find

\displaystyle\int e^x \cos(x) \cosh(x) dx

Try doing it using only the cyclic nature of teh derivatives of the functions in the integrand.

There's a host of different IBP to do, although ultimately you end up doing the same things whichever parts you take.

Reply 2790

henpen

Original post by Indeterminate

Problem 458***

For

u, v >0

show that

\displaystyle \int^{\infty}_{-\infty} \dfrac{u\cos vx - x\sin vx}{(u^2+x^2)^2} dx = \dfrac{\pi}{2u^2e^{uv}}.

This is trivialish via complex analysis, just use de Moivre and a semicircular contour.

Reply 2791

user2020user

16

Problem 461 **/***

Evaluate

\displaystyle \int \dfrac{\log(1+x)}{x^2 + 1} \text{ d}x

(edited 10 years ago)

Reply 2792

Lord of the Flies

19

Original post by Khallil

Problem 461 **/***

Presumably over

[0,1]

. In any case, this has been posted already (and something very similar to 458 was posted as well). I think the thread needs a break from integration :biggrin:

Reply 2793

user2020user

16

Original post by Lord of the Flies

Presumably over

[0,1]

. In any case, this has been posted already (and something very similar to 458 was posted as well). I think the thread needs a break from integration :biggrin:

I had a go at the indefinite integration but kept going around in circles. All of my substitutions kept spiraling me around to the same form :tongue:

Also, I've got a quick question. For integrals with lower and upper bounds, say

\displaystyle \int_{a}^{b}

, are we integrating over the interval

[a, b]

or

(a, b)

?

(edited 10 years ago)

Reply 2794

Flauta

9

Original post by Khallil

Problem 461 **/***

Evaluate

\displaystyle \int \dfrac{\log(1+x)}{x^2 + 1} \text{ d}x

Love that problem, though I'm sure I've seen it on this thread before? At any rate I'll leave it for someone who's not done it before

Reply 2795

rayquaza17

17

Problem 462 ***

A is a symmetric matrix with eigenvectors u and v and with corresponding, distinct eigenvalues λ, μ. Show that the eigenvectors are orthogonal.

(Sorry if this question has been posted before, I don't want to look through 461 other questions :L, also I'm sorry if it's too easy/too hard/no one has any idea what I'm on about)

Reply 2796

user2020user

16

Original post by Flauta

Love that problem, though I'm sure I've seen it on this thread before? At any rate I'll leave it for someone who's not done it before

The indefinite integral? :eek:

Reply 2797

Lord of the Flies

19

Original post by Khallil

I had a go at the indefinite integration but kept going around in circles. All of my substitutions kept spiraling me around to the same form :tongue:

Ah, definitely no closed form for this one.

Also, I've got a quick question. For integrals with lower and upper bounds, say

\displaystyle \int_{a}^{b}

, are we integrating over the interval

[a, b]

or

(a, b)

?

The Riemann integral is over

[a,b]

(in the sense that it requires

f

to be defined everywhere in the interval of integration). Improper integrals are then defined as limits.

Reply 2798

Flauta

9

Original post by Khallil

The indefinite integral? :eek:

No of course not :tongue:

But the definite integral from 0 to 1 only needs a substitution to evaluate

(Will admit that I misread the question and thought it said the definite integral from 0 to 1)

(edited 10 years ago)

Reply 2799

Smaug123

15

Original post by Khallil

I had a go at the indefinite integration but kept going around in circles. All of my substitutions kept spiraling me around to the same form :tongue:

Also, I've got a quick question. For integrals with lower and upper bounds, say

\displaystyle \int_{a}^{b}

, are we integrating over the interval

[a, b]

or

(a, b)

?

Notice that for nice enough integrands, they're the same, because individual points as discontinuities don't alter the value of the integral.

Original post by rayquaza17

Problem 462 ***

A is a symmetric matrix with eigenvectors u and v and with corresponding, distinct eigenvalues λ, μ. Show that the eigenvectors are orthogonal.

(Sorry if this question has been posted before, I don't want to look through 461 other questions :L, also I'm sorry if it's too easy/too hard/no one has any idea what I'm on about)

Yay

Spoiler

The Proof is Trivial!

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