#### 6.53 arcsin integral (18.3.97)

##### 6.53.1 Bernard Marcheterre

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?

##### 6.53.2 Robert Mc Dougall (27.3.97)

We both reached the same limit :o) A package written with the all the asumptions made for all trigonometric functions would be awesome .

##### 6.53.3 David Holmgren (4.4.97)

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.

##### 6.53.4 Giorgio Taricco (11.4.97)

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).