Question

$\int _ { - 1 } ^ { 1 } [ x + [ x + [ x ] ] ] d x$ where [•] denote greatest integer function, is equal to -

• A. – 2
• B. – 1
• C. – 3
• D. – 4

Answer 1

Answer: (C)

– 3

Answer 2

I = $\int _ { - 1 } ^ { 0 } [ x + [ x + [ x ] ] ] d x$ + $\int _ { 0 } ^ { 1 } [ x + [ x + [ x ] ] ] d x$

let I₁ = $\int _ { - 1 } ^ { 0 } [ x + [ x + [ x ] ] ] d x$

–1 £ x < 0 Ž [x] = – 1

Ž [x + [x]] = [x – 1] = [x] – 1 = – 2

Ž [x + [x + [x]]] = [x – 2] = [x] – 2 = – 1 – 2 = – 3

Ž I₁ = $\int _ { - 1 } ^ { 0 } - 3 . d x$ = – 3 = – 3

Similarly I₂ = 0 Ž I₁ + I₂ = – 3