TITLE: Compute $\lim_{n\to\infty} \int_{0}^{\pi/4}\tan^n x \ dx$
QUESTION [6 upvotes]: I'm trying to find some nice proofs for the following limit
$$\lim_{n\to\infty} \int_{0}^{\pi/4}\tan^n x \ dx$$
One way is to use the integration by parts. What else can we do here? Are there some fast ways?
All answers will have my upvote. Thanks!

REPLY [3 votes]: $$I_n = \int_0^{\pi/4} \tan^n(x) dx = \int_0^{\pi/4} \tan^{n-2}(x) \sec^2(x) dx - \int_0^{\pi/4} \tan^{n-2}(x)dx$$
$$I_n + I_{n-2}= \int_0^1 t^{n-2} dt = \dfrac1{n-1}$$ Note that $I_n$ is monotone decreasing with $n$ and is bounded below by $0$ and hence it converges.
Hence, if $\lim_{n \to \infty} I_n = L$, we have that $L + L = \lim_{n \to \infty} \dfrac1{n-1} \implies L = 0$