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Question: \(\int_{0}^{1}{(\{ x\}\lbrack x + 1\rbrack)dx}\) (where {.} denotes fractional part of x and [.] den...

∫01({x}[x+1])dx\int_{0}^{1}{(\{ x\}\lbrack x + 1\rbrack)dx} (where {.} denotes fractional part of x and [.] denotes greatest integer of x) is equal to -

A

1

B

0

C

12\frac{1}{2}

D

None of these

Answer

12\frac{1}{2}

Explanation

Solution

∫01(x–[x])([x]+1)\int_{0}^{1}{(x–\lbrack x\rbrack)(\lbrack x\rbrack + 1)}Ž∫01(x)(1)dx\int_{0}^{1}{(x)(1)dx}

Q 0 < x < 1 Ž [x] = 0 Ž (x2/2)01(x^{2}/2)_{0}^{1} = ½