The value of ∫02x[x2+1]dx, , where [x] is the greatest integer less than or equal to x is:

For x∈[0,2],x2+1∈[1,5] ,we must break [0,2]=[0,1]∪[1,2]∪[2,3]∪[3,2] ∫02x[x2+1]dx=∫01x[x2+1]dx+∫12x[x2+1]dx+∫23x[x2+1]dx+∫32x[x2+1]dx                   =∫01xdx+∫12x2dx+∫23x3dx+∫32x4dx                   =12+13[23/2−1]+14[9−4]+15[32−35/2]                 =46960+1323/2−1535/2