MATLAB: How to get an analytical solution

analytical solutionintegrationnumerical solution

    syms tp t
    NN=((exp(-1.45*10^(-4)*tp^1.274))^1.5)/2 
    z=4/3*pi*NN*(t-tp)^3
    Z=int(z,tp,0,t)
output is
Z =
int((2*pi*exp(-(2674777890687885*tp^(637/500))/18446744073709551616)^(3/2)*(t - tp)^3)/3, tp, 0, t)
that is not an analytical solution, then how can I get the analytical solution?
If t=10 ,how to get the numerical solution?
Thanks for reading.

Best Answer

The existence of an analytical solution is not always necessary. I cannot prove mathematically but it might be the case that this function does not have an analytical solution at all. It might also be the limitation of MATLAB symbolic engine. In the first case, there is not much you can do other than resorting to the numerical solution. In the second case, you may try some other symbolic math engine. As far as the numerical result is concerned, you can get it as with integral().
syms tp t
NN=((exp(-1.45*10^(-4)*tp^1.274))^1.5)/2;
z=4/3*pi*NN*(t-tp)^3;
Z=int(z,tp,0,t);
Z_function = matlabFunction(z);
Z_numerical = integral(@(tp) Z_function(10, tp), 0, 10);
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