Let [ x ] and x be the integer part and fractional part of a real number x respectively. The value of the integral ₀⁵[x] x d x is -

Step-by-step solution

array l ₀⁵[x] x d x= ₀⁵[x](x-[x]) d x= ₀¹ 0 . d x+ ₁² 1 .(x-1) d x+ ₂³ 2(x-2) d x+ ₃⁴ 3(x-3) d x+ ₄⁵ 4(x-4) d x = ( (x-1)² 2 )₁²+2 ( (x-2)² 2 )₂³+3 ( (x-3)² 2 )₃⁴+4 ( (x-4)² 2 )₄⁵ = 1 2 + 2 2 + 3 2 + 4 2 =5 array