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Question: \(\int _ { 1 } ^ { 4 } \{ x \} ^ { [ x ] }\) dx where {.} ®fractional function and [.] ® greatest in...

∫14{x}[x]\int _ { 1 } ^ { 4 } \{ x \} ^ { [ x ] } dx where {.} ®fractional function and [.] ® greatest integer function :

A

1312\frac { 13 } { 12 }

B

12\frac { 1 } { 2 }

C

1512\frac { 15 } { 12 }

D

None

Answer

1312\frac { 13 } { 12 }

Explanation

Solution

I = ∫12(x−1)1\int _ { 1 } ^ { 2 } ( x - 1 ) ^ { 1 } dx + ∫23(x−2)2\int _ { 2 } ^ { 3 } ( x - 2 ) ^ { 2 } + ∫34(x−3)3\int _ { 3 } ^ { 4 } ( x - 3 ) ^ { 3 } dx

I = 1312\frac { 13 } { 12 }