Integral $\int_0^1 \frac{ sin(\pi x)}{x+n }dx$

Question

Let $(u_n)_{n>0}$ defined by $$u_n=\int_0^1 \frac{ \sin \pi x}{x+n}\mathrm dx$$ I want to prove that for all $n\in \mathbb{N}^*$, $(u_n)$ is decreasing.

How can I deduce that $(u_n)$ is convergent, and what is its limit. Thanks for all hints.

Answer 1

We have $0 \le u_n \le \int_0^1 \frac{ 1}{x+n}\mathrm dx \le \int_0^1 \frac{ 1}{n}\mathrm dx = \frac{1}{n}$ for $n \ge 1$.

Hence: $u_n \to 0$.