PROBLEM 3. Consider the sequence with the general term

$$
I_{n}=\int_{0}^{\frac{\pi}{4}} e^{n x}\left(\tan^{n-1} x+\tan^{n} x+\tan^{n+1} x\right) d x, n \geq 1
$$

Calculate $\lim _{n \rightarrow \infty} n\left(\sqrt[n^{2}]{n I_{n}}-1\right)$.
## Problem 3.

We have $I_{n}=\int_{0}^{\frac{\pi}{4}} e^{n x} \operatorname{tg}^{n-1} x\left(1+\operatorname{tg}^{2} x\right) d x+\int_{0}^{\frac{\pi}{4}} e^{n x} \operatorname{tg}^{n} x d x=I_{1}+I_{2}$. The first integral becomes, by integration by parts, $I_{1}=\left.e^{n x}\left(\frac{1}{n} \operatorname{tg}^{n} x\right)\right|_{0} ^{\frac{\pi}{4}}-I_{2}=\frac{1}{n} e^{\frac{n \pi}{4}}-I_{2}(3 p)$. From here $n I_{n}=e^{\frac{n \pi}{4}}$ and the limit to be calculated is $\lim _{n \rightarrow \infty} \frac{e^{\frac{\pi}{4 n}}-1}{\frac{1}{n}}=\frac{\pi}{4}(4 p)$.