Solve with integrating an implicit function

[Helge Gabert]

I recently came, by chance, across this interesting post from one of the hpmuseum archives, originally posted by Valentin Albillo

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Test 7 - Solving a definite integral of an implicit function:
Find X in [0,1] such that

/X
|
| y(x).dx = 1/3
|
/0
where y(x) is an ultra-radical function (a member of the family of elliptic functions) implicitly defined by:
5
y + y = x
using precisions of 1E-3 and 1E-6 for the integral.
HP-71B code:

X=FNROOT(0,1,INTEGRAL(O,FVAR,1E-3,FNROOT(0,1,FVAR^5+FVAR-IVAR))-1/3)
this gives:
X = 0.854136725005 in 433 seconds (precision = 1E-3)
= 0.854138746461 in 771 seconds (precision = 1E-6)

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Obviously, this is very easy on the HP 71B, but it is also no problem for my HP 50G, with the following code

<< -> x
<< 'Y^5+Y=x' 'Y' {0 1} ROOT >>
>>

'FY' STO

<<
<< 0 X 'FY(Z)' 'Z' \int 1 3 / - >>
'X' {0 1} ROOT
>>
EVAL

(and setting Number Format to Fix 3 or Fix 6 in the Mode Menu).

However, I run into problems with the HP Prime

Choosing 'FY' as a CAS program

(y) -> BEGIN
RETURN(nSolve((t^5+t)=y , t));
END;

So far, so good: FY(x), x real, with nSolve gives me the one real solution, as I'm not interested in the four complex ones (which would pop up with, say, fsolve).

But now

nSolve(int(FY(z),z,0,x)-1/3=0,x)

doesn't work, because FY can't take a symbolic argument (i.e., z). Or can it?

Any suggestions? There must be a simple way of doing this on the Prime, but my brain is stuck trying to translate this from HP50G syntax.