Limit $ \lim_{n\to\infty}\int^{n}_{\beta}\frac{\alpha}{u(u^{\alpha}+2+u^{-\alpha})}$

Question

Let ${\alpha}, {\beta} \in \mathbb {R^+}$ For every integer $n>0$ define :

$$a_n=\int^{n}_{\beta}\frac{\alpha}{u(u^{\alpha}+2+u^{-\alpha})}$$

Compute $\displaystyle \lim_{n\to\infty}a_n$

(Answer is terms of $\alpha$ and $\beta$) Please help I don't know what to do here. This was on a test.

Thanks.

Answer 1

Rearranging ie multiplying numerator and denominator by $u^a $ we have $\frac {au^{a-1}}{(u^{2a}+2u^a+1)} $ now let $u^a=k $ thus $au^{a-1}=dk $ thus the integral becomes $\int _{b^a} ^{\infty} \frac {1}{(k+1)^2} $ . Hope you can continue from here. The result is $\frac {1}{b^a+1} $. I have used a,b instead of $\alpha,\beta $