Question

Calculate the following:
The limit as n goes to infinity, n * integral (from 1 to n) of xdx / (n^(3) + [x]^(2) )

Answer 1

\[
g(x)=)\int \frac{x}{n^3+x^2} \, dx=\frac{1}{2} \ln \left(n^3+x^2\right)\\
g(n)-g(1)=\frac{1}{2} \ln \left(n^3+n^2\right)-\frac{1}{2} \ln \left(n^3+1\right)=\\
\frac{1}{2} \ln \left(\frac{n^2 (n+1)}{(n+1) \left(n^2-n+1\right)}\right)=\\
\frac{1}{2} \ln \left(\frac{n^2 }{ n^2-n+1 }\right)\\
\text { Now}\\

\lim_{n->\infty}\frac{1}{2}n \ln \left(\frac{n^2 }{ n^2-n+1 }\right)=\frac 1 2

\]
For the last limit use L'Hospital's Rule