IIT JEE , AIEEE Forum - Mathematics , Integral Calculus

arvind1990

integrate f(x)dx from 0 to n where n is a +ve integer and f(x) is GREATEST INTEGER FUNCTION.
Can anyone xplain me in detail(with all steps)

KAB

_{You must break the integrals.So}

_{[ 0]}

^{[n ]} [x]dx=_{[0 ]}

^{[1 ]} (0)dx+_{[ 1]}

^{[2 ]} (1)dx+........+_{[[x] ]}

^{[x ]} [x]dx

puneet

hiiiiiii

well KAB has told the rite approach .. lemme complete the question ..

₀ⁿ [x]dx=₀

¹ (0)dx+₁

²(1)dx+........+ _n-1

^[n (n-1)dx

I hope u are clear how this can be broken .. rite ..

Now, ₀ⁿ [x]dx=₀

¹ (0)dx+₁

²(1)dx+........+ _n-1

^[n (n-1)dx

= 0 + 1.1 + 2.1 + 3.1 + .... + (n-1).1

= 1 + 2 + 3 + ..... + (n-1)

= n(n-1)/2

I hope the answer is clear ..

cheers