Let [x] be the smallest integer less than or equal to x, and let {x} be the decimal part of x, ie {x}=x-[x]. For any integer n>1, evaluate the following integral from 1 to n, in terms of n:
[x]*{x}*x dx
Assuming that k is an integer, the integrand in the subinterval k to (k+1) is equal to kx(x-k) = k*x^2 – k^2*x.
Integrating this in the said subinterval w.r.t x, we obtain:
k((k+1)^3 – k^3)/3 – k^2*((k+1)^2 – k^2)/2
= k^2/2 + k/3 (upon simplification)
= f(k) (say)
Then, the required definite integral will be given by:
f(1) + f(2) + ……+ f(n-1)
= (1/2)*Sum (k=1 to n-1) (k^2) + (1/3)*Sum (k=1 to n-1) (k)
= (n(n-1)/12)*(2n+1)
= n(n-1)(2n+1)/12
Q E D
Edited on July 14, 2008, 2:32 pm
Solution