# How to perform operation including multiple NIntegrate?

Let $$f$$ be some interpolated function (so not easily integrable in a strict way), and $$g$$ is analytic function (for example $$\exp$$). I want to perform indefinite integral, and plug it into $$g$$, and finally perform definite integral.

NIntegrate[g[NIntegrate[f[x],{x,0,y}]],{y,0,1}]


However, it fails since NIntegrate only allow definite integral. If I substitute Integrate for the second NIntegrate, it works, but it takes very long time, in my impression. I want to know how to perform this integral and obtain interpolated function $$\int_0^x dx f(x)$$. Since we can easily differentiate interpolated function directly, I think it should be possible.

Alternatively, if $$g=1$$ we can rewrite it as a multiple integral,

NIntegrate[Boole[x<y]f[x],{x,0,1},{y,0,1}]


and it works well with much less calculation time. So I also want to generalize such a kind of approach to the case with general $$g$$ if possible.

The integration NIntegrate[g[NIntegrate[f[x],{x,0,y}]],{y,0,1}]fails because the inner NIntegrate doesn't know about y to be numeric!
int[y_?NumericQ]:=NIntegrate[f[x],{x,0,y}]