Calculate $\lim _{n\rightarrow +\infty} \sum _{k=0}^{n} \frac{\sqrt{2n^2+kn-k^2}}{n^2}$

Hint:

Use the substitution:

$$u = \frac{3}{2} \sin(t)$$

Hint. Just make the change of variable $$ u=\frac 32 \sin x,\qquad du = \frac 32 \cos x \,dx, $$ giving $$ \int^{\frac{1}{2}}_{-\frac{1}{2}} \sqrt{\frac{9}{4}-u^2} _,du=\frac 94\int^{\arcsin\frac{1}{3}}_{-\arcsin\frac{1}{3}} \cos^2 x \,dx=\frac 98\int^{\arcsin\frac{1}{3}}_{-\arcsin\frac{1}{3}}\left(1+ \cos 2 x\right) dx. $$ The conclusion is then standard.