For any integer n greater than 1, evaluate this definite integral:
n
∫ ⌊x⌋*x2*{x3} dx
0
Note: ⌊x⌋ is defined as the floor of x, which is equal to the greatest integer less than or equal to x, and:
{x} = x - ⌊x⌋
We divide the integral into n separate integrals using limits:
(0,1) (1,2) ... (n-1,n). Then we sum the results. x indicates floor(x)
int( x x^2 (x-x)^3 dx, x=0 to n
= sum [ int i x^2 (x-i)^3 dx, x=i to i+1], i= 0 to n-1
= sum [(1/60) (15 i^3 + 24 i^2 + 10 i)], i=0 to n-1
= (1/240) (15 n^4 + 2 n^3 -13 n^2 - 4 n)
Edited on September 27, 2024, 9:47 am
soln
