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How to NDSolve a set of equations, one of which itself contains NIntegrate of a desired function waited to be solved by NDSolve first? For example,

f[x_]:=NIntegrate[g[x0],{x0,0,x}]; NDSolve[{f'[x]==g[x],g'[x]==f[x],f[0]==0,g[0]==0},{f,g},{x,0,1}];

You can construct many specific examples like this one, mayebe my question are too general.

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Why not

{sol} = NDSolve[{f'[x] == g[x], g'[x] == f[x], f[0] == 1, g[0] == 0}, {f, g}, {x, 0, 1}]

omitting the definition of f?

Check:

{Inactivate@NIntegrate[g[x0], {x0, 0, x}], f[x]} /. sol /. x -> 0.5 // Activate (* {0.521095, 0.521095} *)
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