How can we prove $\int_1^\pi x \cos(\frac1{x}) dx<4$ by hand? Is there any way we can prove this definite integral inequality by hand:
$$
\int_{1}^{\pi}x\cos\left(1 \over x\right)\,{\rm d}x < 4
$$ 
I don't where to start even, please help. That
$\displaystyle\cos\left(1 \over x\right)\ \leq\ 1$ doesn't seem to help because $\displaystyle\int_{1}^{\pi}x\,{\rm d}x\ >\ 4$.

The integral of x dx is only a little bit greater than 4. 
For x < pi, cos (1/x) is not just ≤ 1. It's ≤ cos (1/pi). For x ≤ 2, it's ≤ cos (1/2). 
