Integrating 1/(1+tan(x)^e) from 0 to pi/2: Challenges and Possible Approaches

[iceblits]

Integrate(1/(1 + [ tan x ] ^e)) from 0 to pi/2

I don't think there's an anti derivative so some other method has to be used to get an exact answer (no approximation). I've tried using Taylor series and Eulers formula. Any help would be great..maybe if you could just point me in the right direction even...

 

[paulfr]

Is that denominator ...
1 + [ tan x^e ]
or
1 + [ tan x ] ^e ?

 

[iceblits]

its 1 + [ tan x ] ^e

 

[iceblits]

Ok I think I got it..I think its pi/4.

 

[disregardthat]

Split the integral up in two parts, from 0 to pi/4 and pi/4 to pi/2. Then use the substitution u = pi/2-x on the second part. Then you will see that it is much easier to compute. How did you arrive upon pi/4? I'm curious.

 

[iceblits]

i just used rectangle approximation and it seemed that it would equal pi/4 (i knew that it had to be an exact answer because i was told so)..your way is probably going to work nicely as well..i will try it..

 

[iceblits]

Ok I tried that and I don't really see where its going. The integral doesn't seem to be computable by finding an anti derivative

 

[Ray Vickson]

Integrate(1/(1 + [ tan x ] ^e)) from 0 to pi/2

I don't think there's an anti derivative so some other method has to be used to get an exact answer (no approximation). I've tried using Taylor series and Eulers formula. Any help would be great..maybe if you could just point me in the right direction even...

What is e? Is it the standard base of natural logarithms, or just some other (positive?) number?
Using Maple I have found that for e = integer 1,...,10, the integral has the same value for all such e, and (by numerical integration), has the same value also for non-integer e. However, at present I don't understand why this happens. You could use Wolfram Alpha to verify this for yourself.

RGV

 

[iceblits]

woah..interesting...I will check that out and yes its e as in the 2.71828183...

 

[iceblits]

wowwwww..thats so cool.. i noticed that before but i didnt think it was the exact same value (i thought it might be off by a few decimal points or something because the region was so small, but using mathematica i can confirm what u said)

 

[iceblits]

I think its because tangent is a relationship between the sin and the cos so the ratio levels out

 

[iceblits]

edit: nevermind i don't know why it is yet..but I intend to find out :)

 

[iceblits]

ok the reason y it does that is because 1/(1+tan(x)^n) is has integral zero over the line y=-pi/2(x)+1