Using the result from DSolve is not intuitive for beginners. Try this:
f0[y_] := ((Cosh[a0 t] - Cosh[y t]) Csch[a0 t])/t;
sol = DSolve[{f1'''[y] -
f1'[y] Cos[\[Alpha]] + (3 f0'[y])/(2 f0[y]) == 0, f1'[0] == 0,
f1[a0] + a1 f0'[a0] == 0, f1'[a0] + a1 f0''[a0] == 0}, f1[y], y];
int1 = Integrate[(f0[y])^2 (f1[y] /. sol[[1]]), {y, 0, a0}]
or look up the documentation for DSolveValue.
