Year 12s: what will be the first way you research your future options? >>

Year 12s: what will be the first way you research your future options? >>

Year 12s: what will be the first way you research your future options? >>

Interesting problem.

\lim_{a\to\infty} \displaystyle\int_{0}^{a} \dfrac{sinx}{x}dx

It has a cool answer.

Original post by Louisb19

\lim_{a\to\infty} \displaystyle\int_{0}^{a} \dfrac{sinx}{x}dx

It has a cool answer.

Off the top of my head, I can think of two ways to do this:

DUTIS:

Spoiler

Laplace transforms:

Spoiler

OP

Original post by Zacken

Off the top of my head, I can think of two ways to do this:

DUTIS:

Spoiler

Laplace transforms:

Spoiler

It's a pretty cool answer isn't it.

Original post by Louisb19

It's a pretty cool answer isn't it.

Indeed, can you hazard a guess as to what

\displaystyle \int_{-\infty}^{\infty} \frac{\sin x}{x} \, \mathrm{d}x

is?

OP

Original post by Zacken

Indeed, can you hazard a guess as to what

\displaystyle \int_{-\infty}^{\infty} \frac{\sin x}{x} \, \mathrm{d}x

is?

I'd guess since

f(x) = \dfrac{\sin x}{x} = f(-x) = \dfrac{\sin (-x)}{-x}

(I think?)

It would be

2\displaystyle \int_{0}^{\infty} \frac{\sin x}{x} dx = \pi

Original post by Louisb19

I'd guess since

f(x) = \dfrac{\sin x}{x} = f(-x) = \dfrac{\sin (-x)}{-x}

(I think?)

It would be

2\displaystyle \int_{0}^{\infty} \frac{\sin x}{x} dx = \pi

Precisely.

\sin (-x) = -\sin x

so

f(x)

is an even function.

So how about

\displaystyle \int_{0}^{\infty} \frac{\sin^2 x}{x^2} \, \mathrm{d}x

?

Original post by Zacken

Off the top of my head, I can think of two ways to do this:

DUTIS:

Spoiler

Laplace transforms:

Spoiler

the Laplace approach is not quite correct

Original post by TeeEm

the Laplace approach is not quite correct

Is this correct (sorry, my Laplace is horridly rusty):

\displaystyle L\left\{\frac{\sin x}{x}\right\} = \int_0^{\infty} L\{\sin t\}\, \mathrm{d}s = \int_0^{\infty} \frac{1}{s^2 + 1} \, \mathrm{d}s = \frac{\pi}{2}

(edited 8 years ago)

Original post by Gregorius

So how about

\displaystyle \int_{0}^{\infty} \frac{\sin^2 x}{x^2} \, \mathrm{d}x

?

Is this just using

Spoiler

Original post by Louisb19

\lim_{a\to\infty} \displaystyle\int_{0}^{a} \dfrac{sinx}{x}dx

It has a cool answer.

You might like this. You come up with quite a few interesting problems, so you can post them there in the future! :smile:

:smile:

Original post by Zacken

Is this correct (sorry, my Laplace is horridly rusty):

\displaystyle L\left\{\frac{\sin x}{x}\right\} = \int_0^{\infty} L\{\sin t\}\, \mathrm{d}s = \int_0^{\infty} \frac{1}{s^2 + 1} \, \mathrm{d}s = \frac{\pi}{2}

I just got back ...
food first!
Then I will write and post the method

Original post by TeeEm

I just got back ...
food first!
Then I will write and post the method

Bon appetit! :smile:

:smile:

Original post by Zacken

Bon appetit! :smile:

:smile:

here is the one and another 3 for the interest of whoever cares

Attachment not found

Attachment not found

edexcel sending.jpg

I posted earlier another integral problem here
http://www.thestudentroom.co.uk/showthread.php?t=3327325

Maybe this is easy to be done now

Original post by TeeEm

here is the one and another 3 for the interest of whoever cares

Attachment not found

Attachment not found

edexcel sending.jpg

I posted earlier another integral problem here
http://www.thestudentroom.co.uk/showthread.php?t=3327325

Maybe this is easy to be done now

Many thanks!

Original post by Zacken

Is this just using

Spoiler

Looks about right; what is amusing in the context of the original question is the actual value of the answer...

Unfortunately the pattern does not continue for

\displaystyle \int_{0}^{\infty} \frac{\sin^n x}{x^n} \, \mathrm{d}x

for n in general...

(edited 8 years ago)

Original post by Gregorius

Looks about right; what is amusing in the context of the original question is the actual value of the answer...Unfortunately the pattern does not continue for

\displaystyle \int_{0}^{\infty} \frac{\sin^n x}{x^n} \, \mathrm{d}x

for n in general...

I was just about to ask about that. :lol:

:lol:

Interestingly enough we do have that becomes

\frac{3\pi}{8}

(edited 8 years ago)

Original post by Gregorius

Looks about right; what is amusing in the context of the original question is the actual value of the answer...

Unfortunately the pattern does not continue for

\displaystyle \int_{0}^{\infty} \frac{\sin^n x}{x^n} \, \mathrm{d}x

for n in general...

I'll be back in half an hour or so, I'm going to play around with that integral and see if I can any nice recurrences or patterns! Thanks for this!

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