KVPY SX SB Exam 2012 Question Paper

Question : 14 of 120

Marks: +1, -0

Let [x] and {x} be the integer part and fractional part of a real number x respectively. The value of the integral

∫↖{5}↙{0}[x ] \{x\} dx

Solution:

∫↖{5}↙{0}[x ] \{x\} dx

=∫↖{5}↙{0}[x](x-[x])dx

=∫↖{1}↙{0}d.dx+∫↖{2}↙{1}1 .(x-1)dx

+∫↖{3}↙{2}2(x-2)dx+∫↖{4}↙{3}3(x-3)dx

+∫↖{5}↙{4}4(x-4)dx

=((x-1)^2/2)^2_1

+2((x-2)^2/2)^3_2

+3((x-3)^2/2)^4_3

+4((x-2)^2/2)^5_4

=1/2+2/2+3/2+4/2=5

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