Re-integration problem of abstract function

[yangyongkang]

Recently I tried to solve the re-integration of some abstract functions with xcas and found some problems.for example:

Code:

diff(integrate(integrate(f(sqrt(x^2+y^2)),y,-(sqrt(t^2-x^2)),sqrt(t^2-x^2)),x,-t,t),t)

The answer we want is

Code:


2pi*diff(f(t),t)*t

But the answer given by XCAS

Code:


"Integral in limit not implemented yet Error: Bad Argument Value"

Related issues

Code:

∫(∫(e^(-(x^2+y^2)/2),y,-(sqrt(a^2-x^2)),sqrt(a^2-x^2)),x,-a,a)

The answer from XCAS

Code:

integrate(sqrt(pi)*1/(sqrt(2))*erf(sqrt(a^2-x^2)*exp(ln(2)/2)/2)*exp(-x^2/2)-sqrt(pi)*1/(sqrt(2))*erf(-sqrt(a^2-x^2)*exp(ln(2)/2)/2)*exp(-x^2/2),x,-a,a)

In fact, we can get the correct answer through polar transformation.

Code:

∫(∫(e^(-r^2/2)*r,r,0,a),x,0,2π)

Get the answer

Code:

2*pi*(-exp(-a^2/2)+1)

Looking forward to the update of hp prime firmware in 2019