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Topic: Integral calculation!
Replies: 2   Last Post: Mar 30, 2013 10:42 PM

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William Elliot

Posts: 1,522
Registered: 1/8/12
Re: Integral calculation!
Posted: Mar 30, 2013 10:42 PM
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On Sat, 30 Mar 2013, José Carlos Santos wrote:
> On 30/03/2013 09:32, Ray Vickson wrote:
>

> > > I need to calculate integral(0,pi/4) x.tan x dx.
> > > http://at.yorku.ca/cgi-bin/bbqa?forum=calculus;task=list

> >
> > Maple gets
> > -(1/8)*pi*ln(2) + (1/2)*Catalan,
> > where Catalan is Catalan's constant, equal to
> > Catalan = sum{(-1)^n 1/(2n+1)^2,n=0..infinity} = 0.9159655942.
> >
> > Numerically, the integral is about 0.1857845357,

>
> Mathematica gets the same thing. Of course, this suggests that x.tan(x)
> doesn't have an elementary primitive.


I agree and think it all starts with

integral(0,pi/2) x/sin x * dx = 2.sum(n=1,oo) 1/(2n-1)^2 = k

From that I thought to get integral(0,pi/2) x/cos x * dx
integral(0,pi/2) x/sin x * dx
. . = integral(pi/2,0) (pi/2 - x)/sin(pi/2 - x)
. . = integral(0,pi/2) (x - pi/2)/cos x
. . = integral(0,pi/2) x/cos x - pi/2 * integral(0,pi/2) sec x dx
. . = integral(0,pi/2) x/cos x - pi/2 * sec x tan x|_0^pi/2
. . = integral(0,pi/2) x/cos x - pi/2 * sin x/cos^2 x|_0^pi/2

But failed. If integral(0,pi/2) x/cos x * dx is known can one
use integration by parts for integral(0,pi/2) x/cos x * sin x dx
to make the conclusion?

--
integral dfg = integral f.dg + integral g.df

lim(x->pi/2) x/cot x = lim(x->pi/2) -1/csc^2 x
. . = lim(x->pi/2) -sin^2 x = -1

x.tan x|_0^pi/2 = -pi/2

----



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