# How to evaluate the limit of this defined integral?

Having a hard time finding this limit, it must be some natural log replacement I figured but I quite don't know how to replace it as such

$$\lim _{n\to \infty }\left(n\int _0^1\:\left(\frac{x^n}{\left(x^n+x+1\right)}dx\right)\right)$$

Managed to see that it can be written as almost the natural log by doing this thing here:

$$\lim _{n\to \infty }\left(\int _0^1\left(\frac{\left(nx^{n-1}+1-1\right)}{\left(x^n+x+1\right)}dx\right)\:\:\right)$$

$$\lim _{n\to \infty }\left(\left(ln\left(3\right)-\int _0^1\left(\frac{1}{\left(x^n+x+1\right)}dx\right)\right)\:\:\right)$$

Still have to solve this one: $$\int _0^1\left(\frac{1}{\left(x^n+x+1\right)}dx\right)$$

• I think so there is mistake in 3rd line. Oct 31, 2021 at 19:14
• @RAHUL it was, ln(3) Oct 31, 2021 at 19:19
• Yaa, so that integrand is only left to solve Oct 31, 2021 at 19:22
• You can remove the $n$ at the front and bring it in. Note that as $n\to\infty$, $nx^{n}=nx^{n-1}$ Oct 31, 2021 at 19:26
• $$\int_{0}^{1}\frac{dx}{x^{n}+x+1}$$ converges to $0.69..$ as $n$ approach $\infty$ Oct 31, 2021 at 19:34