How to solve this improper integral?

02082015, 11:29 AM
(This post was last modified: 02082015 11:33 AM by salvomic.)
Post: #1




How to solve this improper integral?
hi,
I need to solve this improper integral \[ \int_{0}^{\infty}{\frac{x^{a}}{x^{2}+1} dx} \] with 1 < a < 1 I try to use limit r>\infty and bound integral from 0 to r, but it give the symbolic form of the integral, without calculation... The integral should be \( \frac{\frac{pi}{2}}{cos(\frac{pi*a}{2})} \) Is it possible in Prime? Thanks! Salvo ∫aL√0mic (IT9CLU) :: HP Prime 50g 41CX 71b 42s 39s 35s 12C 15C  DM42, DM41X  WP34s Prime Soft. Lib 

02082015, 02:58 PM
Post: #2




RE: How to solve this improper integral?
No (at least not yet!).


02082015, 03:48 PM
Post: #3




RE: How to solve this improper integral?
(02082015 02:58 PM)parisse Wrote: No (at least not yet!). thank you for info! I tried to solve it giving some arbitrary values: ex.: a=½ I get in Terminal "Rational unvariate representation is not certified, set proba_epsilon:=0 to certify...", then "Unable to handle singularities..." If I put epsilon to 0 Terminal says "warning, solution does not seem to cancel c_4^2taylorx1", then the same error... a=1/9 I get infinity a=¼ Terminal says "Warning Algebraic extension not implemented yet for poly [1,0,2312,39304,167042]" (repeated with other values), then the result (½)*pi*\sqrt(\sqrt(2)*2+4) a=⅓ the result is pi/\sqrt(3) and in fact those are not the solution Ok, I mustn't oblige the Prime to give more ;) Salvo ∫aL√0mic (IT9CLU) :: HP Prime 50g 41CX 71b 42s 39s 35s 12C 15C  DM42, DM41X  WP34s Prime Soft. Lib 

02092015, 08:10 AM
Post: #4




RE: How to solve this improper integral?
I get errors with Xcas:
for 1/2 "Unable to handle singularities of 2*((sqrt(2)*ln(xsqrt(2)*sqrt(x)+1))/8+(sqrt(2)*atan((sqrt(x)(sqrt(2))/2)/sqrt(2)*2))/4(sqrt(2)*ln(x+sqrt(2)*sqrt(x)+1))/8+(sqrt(2)*atan((sqrt(x)+(sqrt(2))/2)/sqrt(2)*2))/4) at [undef]" I did not try on the emulator, but error handling is poor on the Prime, perhaps it is not returned to the user. You can get the value of the integral by adding int(x^a/(1+x^2),x,inf,0), replace inf and inf by R large, add the upper halfcircle integral of radius R and apply residue theorem, then take the limit as R>inf. You can check for a=1/2 or a=1/3 like this assume(t>0); simplify(eval(subst('int(x^(1/3)/(x^2+1),x,0,inf)',x=t^3))) 

02092015, 08:33 AM
Post: #5




RE: How to solve this improper integral?
(02092015 08:10 AM)parisse Wrote: ... yes, thank you! ir works well. I'm trying to memorize this method, it's so useful. Have a nice day ∫aL√0mic (IT9CLU) :: HP Prime 50g 41CX 71b 42s 39s 35s 12C 15C  DM42, DM41X  WP34s Prime Soft. Lib 

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