This question refers to the integration of the square of arcsin(x). If you give Maple these commands:
>assume(x>0,x<1): >int(arcsin(x),x);
It will respond with the correct result, at least according to Schaum’s Mathematical Handbook of Formulas and Tables, equation 14.471. If, instead you go with:
>assume(x>0,x<1): >int((arcsin(x))^2,x);
The unevalueted integral is returned. I found a way to reach the result (equation 14.476, Schaum) trough intparts from the student package but I wonder, is there a special command that I need to know? Have I reached the limit of the integration kernel with this special case?
We both reached the same limit :o) A package written with the all the asumptions made for all trigonometric functions would be awesome .
When I saw this posting, I thought I might try MuPAD 1.3 on the problem (I realise that this is rather naughty, but not to worry). In the following, asin(x) is the same as arcsin(x):
>> int(asin(x)^2,x)
which after only a short time ( 30 sec) produces:
2 2 1/2 -2x + x asin(x) + 2 asin(x) (-x + 1)
I have no idea of what assumptions MuPAD makes with this problem.
There is a simple way to tell Maple how to calculate
int(arcsin(x)^n,x).
The following code does the job:
> restart:with(student): # Restart and load student package > assume(cos(u)>0): # used later on > Int(arcsin(x)^2,x): # arcsin integral (inert form) > changevar(arcsin(x)=u,%,u): # change variables x->u > value(simplify(%)): # calculate integral > subs(u=arcsin(x),%): # change u back to x > simplify(%); # final simplification
and works for any positive integer power of arcsin(x).