The hard integral thread.

Post some of your tough integrals for us, please.

Show that, for $\alpha \ge 0$,

$\displaystyle \int_0^\infty \frac{(1-x^2)\cos \alpha x}{(1+x^2)^2} \ dx = \frac{\pi \alpha e^{-\alpha}}{2}$

Evaluate $\displaystyle \int_1^\infty \frac{\ln^3 x}{x^2(x-1)} \ dx$

Show that, for $\alpha \ge 0$,

$\displaystyle \int_0^\infty \frac{(1-x^2)\cos \alpha x}{(1+x^2)^2} \ dx = \frac{\pi \alpha e^{-\alpha}}{2}$

A little messy, but ya'll might like it:
$\displaystyle\int \dfrac{1}{1+x^4} \ dx$.

i got shouted at when I suggested someone do that last year! Here's one that's far more reasonable:

$\displaystyle\int \sqrt{\tan{x}}\ dx$.

NO. NO I am not going to show that. But seriously before I try it tmmrw can it be done with a level FM and maths techniques?

I will try these 2 but at present I am trying that of post 69 which has seriously dented my confidence.

I am writing a neat solution for it and I am on the 4th page having done partial fractions 3 or 4 times so far.

The integrals in this thread are getting so ridiculous one would think there is a prize to be won.

Post 82 should be right up your street. Post 83 depends on a trick that you may or may not have seen, though if I had posed it by saying "Show that blah = blah", the value may have given the game away. (In fact, I spent some time making up a tricky question earlier with you in mind based on exchange of order of integration - I was very pleased with the result, until I noticed that it could be done in 10 seconds flat by a couple of Laplace transforms :-( )

Yes, I had a go at post 69 yesterday and it blew me away. Couldn't see any way into it - maybe some contour integral approach?

As for ridiculous integrals: as long as people eventually post solutions, it's a good way of demonstrating unusual techniques, that aren't often seen - useful for all the young bucks going up for their Oxbridge interviews.

NO. NO I am not going to show that.

But seriously before I try it tmmrw can it be done with a level FM and maths techniques?

Q69 is DUTIS twice and it is very long

82 I can only thing complex integration, (from the answer)

83 as you said the lack of answer makes it hard

Not for me then. I do most of the stuff that I post here in my head/on scraps of paper, and I get bored quickly if I have to do lots of tedious algebra. I'll look at it again another time to see if there's a nicer approach.



No, not complex integration (necessarily). Hint:


Hint:

I gave you a direct order to show that result, young fellow-me-lad, and I expect it done by the time I get up tomorrow!!!



If you can, then Kipling's description of a certain Mr Din would be applicable.

i got shouted at when I suggested someone do that last year! Here's one that's far more reasonable:

$\displaystyle\int \sqrt{\tan{x}}\ dx$.

i got shouted at when I suggested someone do that last year! Here's one that's far more reasonable:

$\displaystyle\int \sqrt{\tan{x}}\ dx$.

Mr Fourier should not be neglected at times like this

i got shouted at when I suggested someone do that last year! Here's one that's far more reasonable:

$\displaystyle\int \sqrt{\tan{x}}\ dx$.

That just seems to end up as an even more unpleasant bunch of partial fractions, no?