How can I plot a graph of an integral?

g[x_]=Gamma[x+1]/2-Gamma[x-1]/2 // FunctionExpand

(-(1/2) + 1/2 (-1 + x) x) Gamma[-1 + x]

f[y_] := NIntegrate[g[x], {x, 2, y}]

Plot[f[x], {x, 2, 10}]

enter image description here

You could use NDSolveValue to integrate the function:

int = NDSolveValue[
    {
    f'[x] == Gamma[x+1]/2-Gamma[x-1]/2,
    f[2] == 0
    },
    f,
    {x, 2, 10}
];

Visualization:

Plot[int[x], {x, 2, 10}]

enter image description here

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