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Well, could you start with the infinite series for cosine x?

You'd have (1/x) (-x^2/2! + x^4/4! .....)

\[ \int \frac{\cos x}{x}dx - \int \frac{1}{x}dx \]

Let's see... \[\sum_{n=0}^{\infty} (-1)^n \frac{x^{2n+1}}{(2n+1)!}, \text{for all x = } \mathbb{R}\]

That's the MacLaurin equivalent for cosine

\[ \frac{ \cos x}{x} = \frac{1}{x} \sum_{n=0}^\infty (-1)^n \frac{x^{2n}}{(2n)!} \]

Is it? Did I look at the wrong one? (-:

oh yes that's the hyperbolic one! SOrry!

Oh no wait I have the wrong one, I wrote sine by mistake

Arg soo confusing lol

lets both check again then hehe.

yes I believe I have the right one

I thought you couldn't just do 1/x , you'd have to convert that to a series as well, yes?

+c

Anti-derivative power rule?

all has mistake!

where @mahmit2012 ?

all !but it is time to pray i wll come back and show you

after pray !

lol ... where did i make mistake??

Did we do the same @experimentX ? besides that you completed it already.

I think we did.

n=1, n=0

yes but obviously "ALL" is wrong :D

|dw:1343670857318:dw|

*tilting head sideways to read*
:-D

|dw:1343670990609:dw|

|dw:1343671203407:dw|

Well the final answer is probably going to be ln|x| + something, I think...

yes but you didn't allow to separate 1 and cos because of in cosx/x you have a log function.

can't we cancel 1 and start from n=1 instead?

yes but if you start with 1-cosx in num so you wont confront the mistake.

take a look my solution and figure out the mistake.

Better: Substitute it directly and derive the formula in form of a series representation?

yeah ... my case (-1)^n was mistake ..

and for the last this problem is originally has defined to 0 to inf.

|dw:1343671601188:dw|

I understand, thanks @mahmit2012

perhaps this is related to
http://en.wikipedia.org/wiki/Dirichlet_integral

Not exactly but a little bit !

in the series of cosine ... there is 1 ... cancel out 1 ...
it's simply ... shifting n=0 to n=1

actually that needs to be n \(\pm\) 1on the alternating part

@experimentX you mean shifting the index eh?

yep

since the one gets cancelled ... the after that x is divided.

this gives the value of
|dw:1343673440350:dw|

I've never even heard of Laplace Transform until now :3

I haven't learned either of those yet, the section from a chapter of the text on infinite series.