integer nearst to x and less than x will be [x]

so ₀ò^x [x] dx = ₀ò¹ 0 dx + ₁ò² 1 dx + ₂ò³ 2 dx + ..................... + _[x]-1∫^[x] [x]-1 dx + _[x]∫^x [x] dx

now solve it