`lim_(n to oo) sum_(r=1)^(n) ((r^(3))/(r^(4) + n^(4)))` equals to-

Options

A. ln 2
B. `(1)/(2) ln2`
C. `(1)/(3) ln 2`
D. `(1)/(4) ln2`

Answer

4

Explanation

We have
`underset(n to oo)"lim"underset(r=1)overset(r=n) sum (r^(3))/(n^(4)(((r)/(n))^(4)+1))`
`= underset(0)overset(1) int (x^(3).dx)/(x^(4)+ 1)`
`=(1)/(4) (log (1+x ^(4)))_(0)^(1)`
`underset(nto oo)"lim" underset(r=1)overset(r=n)sum to int`
`=(1)/(4) log 2-0`
Replace
`(r)/(n) to x rArr (1)/(n) to dx`