So Maple 2019 seems to be unable to evaluate a relatively easy integeral for a positive integer, but when the obvious substitution u=cos(x) is made it is able to evaluate the integral.  Is there some setting I am using that is causing Maple to give up without much effort?  
 

Download walter.mw

Can anyone look at this worksheet, and explain why maple seems to complicate an easily evaluated integral?

Hyper.mw

Integrating a positive definite function (normal distribution) and a >= 0 function (Heaviside) should not return a negative value.

with(Statistics);
X := RandomVariable(Normal(1, sqrt(2.25)));
int(PDF(X, x)*Heaviside(x^7-5*x^4-3*x+1), x = -infinity .. infinity);
                         -0.08507120131

Bug.mw

Is there a way to take the laplacian of 1/r and get the "physics" answer of -4*pi*delta(\vec{r})?

My Maple worksheet is attached below - where I am solving two quantum mechanics problems, trying to find the eigenvalues for certain potentials.  Using a previous post in Mapleprime (https://www.mapleprimes.com/questions/221629-Can-Maple-Find-Solution-To-Eigenvalue),  I saw how to get the lowest eigenvalue.  My question is how using numeric solutions, I can get the higher eigenvalues and corresponding solutions.  Is there a way to specify that e (the eigenvalue) has to be in a certain range.  I tried to specify assumptions (e>0.6), but that didn't work.  I know that for the second problem (using the shooting method) that the 4 lowest eigenvalues are 0.184358, 0.70747, 1.2065, 1.5625.  Thank you very much for you consideration.

ODE_BVP_eigenvalues.mw