TITLE: Finding $\lim_{n \to \infty} \int_1^a \frac{n}{1+x^n} \, dx$
QUESTION [7 upvotes]: I'm trying to find $$ \lim_{n \to \infty}   \int_1^a \dfrac{n}{1+x^n} \, dx  ,$$ where parameter $a>1$.
The limit appears to be $\ln 2$ regardless of the value of $a$.
Some trick is probably needed here, but i don't see it.

REPLY [2 votes]: Hint: Put $$x=tan^{2/n}\theta$$
          $$ dx=\dfrac2ntan^{\frac2n-1}\theta \times sec^2\theta . d\theta$$

Now =>   $$\boxed{I_n=\int_\phi^\psi 2tan^{\frac2n-1}\theta . d\theta}$$
  $\phi,\psi$ are the limits accordingly.
  $$\phi=tan^{-1}1=\pi/4$$
  and $$\psi=tan^{-1}[a^\frac n2]$$
  $$Lt_{n\rightarrow\infty}\ \ \ [\psi]=\pi/2$$
$$Lt_{n\rightarrow\infty}[I_n]=\int_{\pi/4}^{\pi/2}\dfrac{2cos\theta}{sin\theta}.d\theta$$.

So,it can be done!