_ n → ∞ ( n ^ 2 ( n ^ 2 + 1)(n + 1) + n ^ 2 ( n ^ 2 + 4)(n + 2) + n...

$\lim_{n \to \infty}\mathop{∑}_{r = 1}^{n}\dfrac{\dfrac{1}{n}}{\left(1 + \dfrac{{r}^{2}}{{n}^{2}}\right)\left(1 + \dfrac{r}{n}\right)} = \mathop{∫}_{0}^{1}\dfrac{dx}{\left(1 + {x}^{2}\right)\left(1 + x\right)}$

Let $I = \mathop{∫}_{0}^{1}\dfrac{dx}{\left(1 + {x}^{2}\right)\left(1 + x\right)}$

Now $x = tan\theta, dx = \sec^{2}\thetad\theta$

$I = \mathop{∫}_{0}^{\dfrac{\pi}{4}}\dfrac{d\theta}{1 + tan\theta} = \dfrac{1}{2}\mathop{∫}_{0}^{\dfrac{\pi}{4}}\dfrac{2cos\thetad\theta}{sin\theta + cos\theta}$

$= \dfrac{1}{2}\mathop{∫}_{0}^{\dfrac{\pi}{4}}\dfrac{\left(sin\theta + cos\theta\right)d\theta}{sin\theta + cos\theta} + \dfrac{1}{2}\mathop{∫}_{0}^{\dfrac{\pi}{4}}\dfrac{\left(cos\theta - sin\theta\right)d\theta}{sin\theta + cos\theta}$

$= \dfrac{1}{2} \times \dfrac{\pi}{4} + \dfrac{1}{2}{\left[\ln \left|\sin\theta + cos\theta\right|\right]}_{0}^{\dfrac{\pi}{4}}$ $= \dfrac{\pi}{8} + \dfrac{1}{4}\ln 2$