The value of `lim_(n to oo) sum_(r=1)^(n)(r^(2))/(r^(3)+n^(3))` is -

Explanation

We write `t_(r )` in the form `t_(r )=(1)/(n)f((r )/(n))`
`t_(r )=(r^(2))/(r^(3)+n^(3))=(1)/(n)*(r^(2)n)/(r^(3)+n^(3))=(1)/(n)*(((r )/(n))^(2))/(1+((r)/(n))^(3))`
Lower limit of `r=1 :. ` lower limit of integration `=lim_(ntooo)(1)/(n)=0`
Upper limit of `r=n :. ` upper limit of integration `=lim_(ntooo)(n)/(n)=1`
Now,
`lim_(ntooo)sum_(r=1)^(n)(r^(2))/(r^(3)+n^(3))=int_(0)^(1)(x^(2))/(1+x^(3))dx`
`=(1)/(3)int_(0)^(1)(3x^(2))/(1+x^(3))dx`
`=(1)/(3)int_(0)^(1)(d(1+x^(3)))/(1+x^(3))`
`=(1)/(3)[ln(1+x^(3))]_(0)^(1)`
`=(1)/(3)[ln2-ln1]`
`=(1)/(3)ln2`