1. ## estimation

I got stucked with this problem !
any rule or thereom to slove it

Find bounds m and M such that $\displaystyle m \le x\sin(x) \le M$
use the to estimate $\displaystyle \int_{0}^{\pi}x\sin(x)dx$

2. If $\displaystyle f(x_1) \leq f(x) \leq f(x_2)$, then $\displaystyle f(x_1)(b-a) \leq \int_a^b f(x) dx \leq f(x_2)(b-a)$

So if you know the boundaries of $\displaystyle m \leq x\sin x \leq M$ on $\displaystyle x \in [0,\pi]$, then you can also estimate the integral with $\displaystyle m(\pi-0) \leq \int_0^\pi x \sin x dx \leq M(\pi-0)$