If [ ] denotes the greatest integer function, then ₁^4 x[x] dx is equal to

Step-by-step solution

The given integral can be split at integer values since [x] changes its value at integers. I = ₁^4 x[x] dx I = ₁^2 x[x] dx + ₂^3 x[x] dx + ₃^4 x[x] dx Substituting the values of [x] in the respective intervals: I = ₁^2 x(1) dx + ₂^3 x(2) dx + ₃^4 x(3) dx I = [ x^2 2 ]₁^2 + [ x^2 ]₂^3 + [ 3x^2 2 ]₃^4 I = ( 4 2 - 1 2 ) + (9 - 4) + ( 48 2 - 27 2 ) I = 3 2 + 5 + 21 2 I = 24 2 + 5 I = 12 + 5 = 17 Answer: 17