A calculus problem by A Former Brilliant Member

$\begin{aligned} \lim_{n \to \infty} n(I_n + I_{n+2}) & = \lim_{n \to \infty} n \left(\int_0^\frac \pi 4 \tan^n x \ dx + \int_0^\frac \pi 4 \tan^{n+2} x \ dx \right) \\ & = \lim_{n \to \infty} n \int_0^\frac \pi 4 \tan^n x (1+\tan^2 x) \ dx \\ & = \lim_{n \to \infty} n \int_0^\frac \pi 4 \tan^n x \sec^2 x \ dx & \small \color{#3D99F6}{\text{Let }t = \tan x \implies dt = \sec^2 x \ dx} \\ & = \lim_{n \to \infty} n \int_0^1 t^n \ dt \\ & = \lim_{n \to \infty} n \cdot \frac {t^{n+1}}{n+1} \bigg|_0^1 \\ & = \lim_{n \to \infty} \frac n{n+1} \\ & = \lim_{n \to \infty} \frac 1{1+\frac 1n} \\ & = \boxed{1} \end{aligned}$