Integral help

TeeEm needing help?!? Thought I'd never see the day! :P

Integration by parts?

happens to me more regularly than I would like to admit ...

However in this question I have done the integral without problems; I just look for a quicker alternative if it exists.
( it is just that I am doing high level maths non-stop since this morning and my brain is now stalling )

I second that idea...and so does Wolfram Alpha 🙂

I split the log

Mathematical lumberjack as well now!

Mathematical lumberjack as well now!

I cut a bit of the work out now, so it is ok...

It is actually one of the questions you posed the other night, that I took and made a question out of ...

Find

INTEGRAL (sin²x/x²) from 0 to infinity, using Laplace transform techniques

This reminds me, Greogrius - I tried computing a recurrence for the integral of nth powers of the since function, i.e a closed form for $\displaystyle \int_0^{\infty} \left(\frac{\sin x}{x}\right)^n \, \mathrm{d}x$ and eventually gave up (although I managed to get the first few nth powers, up to 5 or so) - then a quick math.stackexchange lookup yielded and impressive and complex-looking closed form!

I just borrowed the n=2 case to make a question for my LaplaceTransform resources

This reminds me, Greogrius - I tried computing a recurrence for the integral of nth powers of the since function, i.e a closed form for $\displaystyle \int_0^{\infty} \left(\frac{\sin x}{x}\right)^n \, \mathrm{d}x$ and eventually gave up (although I managed to get the first few nth powers, up to 5 or so) - then a quick math.stackexchange lookup yielded and impressive and complex-looking closed form!

Ooh! Do you have a link to the math.stackexchange article?

Any good method for this?
(please attach I do not have home-works any more!!!)

I have the correct answer (which is part of an undergrad question to which I am writing a solution)
It is just that the method seems a bit too long and I do not want to rewrite the solution as this integral lies somewhere in the middle of a longish problem

Thanks

I saw - I really need to brush up on my Laplace Transforms, perhaps starting with your resources! :-)

Here you go, although most of the information/maths is in the PDFs/papers that they've linked to in the comments. :-)

Excellent! An endless stream of exam questions for the future!

I wrote many questions last week as I am helping 2 students at present,

wait until I update the site as none of them are there

here is the one I made tonight which lead to this integral

$I=\int \log(\frac{x^2+1}{x^2}) \ dx$

Let $x=\tan u \Rightarrow dx=\sec^2 u du$

So:

$I = \int \log(\frac{\sec^2 u}{\tan^2 u}) \ du = \int \log(\frac{1}{\sin^2 u}) \sec^2 u \ du = -2 \int \log(\sin u) \sec^2 u \ du$

Now by parts we have:

$I= -2 \{ \tan u \log(\sin u) - \int \tan u \cot u \ du \} = -2 \{ \tan u \log(\sin u) - u \} + C$

Now $x=\tan u \Rightarrow \sin u = \frac{x}{\sqrt{x^2+1}}$ so:

$I = -2 \{ x \log(\frac{x}{\sqrt{x^2+1}}) - \tan^{-1} x \} + C = x \log(\frac{x^2+1}{x^2}) + 2 \tan^{-1} x + C$

(Confirmed by Wolfram)