# solving this integral using Lebesgue dominated convergence theorem

I'm tried without sucess to solve this integral: $$\lim_{n\to\infty}\int_{0}^{1}(1+x^2)^{-n}dx$$

I know that the why to solve it ,is by using the Lebesgue dominated convergence theorem.

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Hint: look at the function $1/(1+x^2)$. The solution is very straightforward. – Bruno Joyal Feb 11 '13 at 19:57
Yes. Just show that the integrands converge pointwise to ... and that they are dominated by ... – 1015 Feb 11 '13 at 20:00
When asking questions on this site it is customary to include your thoughts about the problem as well as what you have tried to do to solve it. – Antonio Vargas Feb 11 '13 at 20:25
OK, I see that the integrands converge to pi/4 (without the limit) but what about n ? – user61801 Feb 11 '13 at 20:53
First calculate $$\lim_{n \to \infty} (1+x^2)^{-n},$$ then use the dominated convergence theorem to say that $$\lim_{n \to \infty} \int_0^1 (1+x^2)^{-n} \,dx = \int_0^1 \left(\,\lim_{n \to \infty} (1+x^2)^{-n}\right) \,dx.$$ Which part is giving you trouble? – Antonio Vargas Feb 11 '13 at 22:09