Show that $\lim_{n \to \infty} \int_0^{\infty} \frac{1}{1+x^n}~dx = 1$

Break it into two integrals, on $[0,1]$ and on $[1,\infty)$. $$\int_1^\infty\frac{dx}{1+x^n}<\int_1^\infty\frac{dx}{x^n}=\frac1{n-1}$$ so $$\int_1^\infty\frac{dx}{1+x^n}\to0.$$ Also $$1-\int_0^1\frac{dx}{1+x^n}=\int_0^1\frac{x^n}{1+x^n}\,dx <\int_0^1 x^n\,dx=\frac1{n+1}\to0$$ so $$\int_0^1\frac{dx}{1+x^n}\to1.$$