1. Calculate

(i) $\int \frac{\arcsin x}{x^{2}} d x$, for $x \in(0,1)$.

(ii) $\lim _{n \rightarrow \infty}\left(\frac{n}{n^{2}+1^{2}}+\frac{n}{n^{2}+2^{2}}+\ldots+\frac{n}{n^{2}+n^{2}}\right)$.
(i) We integrate by parts and obtain $\int \frac{\arcsin x}{x^{2}} d x=-\frac{1}{x} \arcsin x+\int \frac{1}{x \sqrt{1-x^{2}}} d x$

We perform the variable substitution $\frac{1}{x}=t$

Completion

$$
\begin{aligned}
& \text { (ii) } \sum_{k=1}^{n} \frac{n}{n^{2}+k^{2}}=\frac{1}{n} \sum_{k=1}^{n} \frac{1}{1+\left(\frac{k}{n}\right)^{2}} \\
& \lim _{n \rightarrow \infty} \frac{1}{n} \sum_{k=1}^{n} \frac{1}{1+\left(\frac{k}{n}\right)^{2}}=\int_{0}^{1} \frac{1}{1+x^{2}} d x=\frac{\pi}{4}
\end{aligned}
$$