Rudimental Doctrine with concern to Transformation

The main trick in this question is to use L'Hospital's Rule and Newton-Leibniz Formula.

The given limit is :- $\lim_{n\to\infty} \int_{n}^{2n} \dfrac{n^{3}x}{x^{5}+1}.dx$ It can be written as:- $\lim_{n\to\infty} \dfrac{\int_{n}^{2n} \dfrac{x}{x^{5}+1}.dx}{\dfrac{1}{n^{3}}}$ Now we can apply L'Hospital's Rule and Newton-Leibniz Formula:- $\lim_{n\to\infty} \dfrac{\dfrac{2n\times 2}{(2n)^{5}+1}-\dfrac{n}{n^{5}+1}}{\dfrac{-3}{n^{4}}}$ We can evaluate this limit and it comes out to be $\boxed{\dfrac{7}{24}}$ .

Just for a change

$\displaystyle \int_n^{2n} \dfrac{n^3x}{x^5 + 1} dx = n.\int_1^2 \dfrac{n^3.(nx)}{(nx)^5 + 1} dx$

I call this time saving property(its general substitution), if you divide a factor(here n) in limits replace x with nx & one n outside it.

Our integral reduces to

$\displaystyle \lim_{n \to \infty } \int_1^2 \dfrac{n^5x}{n^5.x^5 +1} dx$

Divide by $n^5$ and applying limit to get

$\displaystyle \int_1^2 \dfrac{1}{x^4} dx = \dfrac{7}{24}$ .