O16-066
> restart;
> f:=n->1/(3+t^2)^n;

                                       1
                         f := n -> ---------
                                         2 n
                                   (3 + t )

> seq(int(f(n),t=0..infinity),n=1..2);

                   1/6 Pi sqrt(3), 1/36 Pi sqrt(3)

> sum(int(f(n),t=0..infinity),n=1..infinity);

                  infinity    infinity
                   -----     /
                    \       |              1
                     )      |          --------- dt
                    /       |                2 n
                   -----   /           (3 + t )
                   n = 1     0

> g:=b->3*x^2+4*y^2+2*b*x-b^2;

                               2      2            2
                  g := b -> 3 x  + 4 y  + 2 b x - b

> with(plots);

  [animate, animate3d, animatecurve, changecoords, complexplot,

        complexplot3d, conformal, contourplot, contourplot3d,

        coordplot, coordplot3d, cylinderplot, densityplot, display,

        display3d, fieldplot, fieldplot3d, gradplot, gradplot3d,

        implicitplot, implicitplot3d, inequal, listcontplot,

        listcontplot3d, listdensityplot, listplot, listplot3d,

        loglogplot, logplot, matrixplot, odeplot, pareto, pointplot,

        pointplot3d, polarplot, polygonplot, polygonplot3d,

        polyhedra_supported, polyhedraplot, replot, rootlocus,

        semilogplot, setoptions, setoptions3d, spacecurve,

        sparsematrixplot, sphereplot, surfdata, textplot, textplot3d,

        tubeplot]

> implicitplot({g(1),g(2)},x=-2..1,y=-2..2);
> 

> # En polaires, r=b/(2+cos(theta)) et I=4b^3/3(I1-4I2+4I3) en passant en polaires puis avec t=tan(theta/2)
> v:=[seq(int(f(n),t=0..infinity),n=1..3)];

       v := [1/6 Pi sqrt(3), 1/36 Pi sqrt(3), 1/144 Pi sqrt(3)]

> 4*b^3/3*(v[1]-4*v[2]+4*v[3]);

                               3
                          1/9 b  Pi sqrt(3)

>