<<Integral depending on parameter >>?

dJ(p)/dp=[0, 1] ∫x/(1+(px)^2)*(x*√(1-x²))dx

=[0, 1] ∫1/(1+(px)^2)*(√(1-x²))dx

=∫1/(1+(px)^2)*(√(1-x²))dx

x=sin(k)

dx=cos(k)

=∫1/1+(p^2)(sink)^2

=∫(coseck)^2/(coseck)^2+p^2

=∫(coseck)^2/(cotk)^2+p^2+1

cotk=t

-(coseck)^2dk=dt

=-∫1/t^2+y^2

where y=sqrt(1+p^2)

-1/yarctan(t/y)

=(-1/sqrt(p^2+1))arctan(t/sqrt(1+p^2))

=(-1/sqrt(p^2+1))arctan((1/x)sqrt((1-x^2)/(1+p^2)))

http://integrals.wolfram.com/index.jsp?expr=1%2F%2...

from 0 to 1 it becomes

pi/(2sqrt(1+p^2))

dJ(p)/dp=pi/(2sqrt(1+p^2))

integrating w.r.t p both sides of equation

J(p)=(pi/2)∫sqrt(1+p^2)

p=tanz

and doing integration

i got the answer to be

(pi/2) ln|sqrt(1+p^2)+p|

am i correct.

sorry,in the beginning i did it as arctan but at the end i just forgot it and took 'a' as a const

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hey Rasa,

J(x,p) ??... is the integral expression not with respect to 'x' ??, how can 'x' be an independent variable of J(x,p) ??

David