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 $\displaystyle\int_{1}^{\pi}x\,{\rm d}x\ >\ 4$.

First of all,
$$∫_{x=1}^π x\cos \frac{1}{x}dx=∫_{t=1}^\frac{1}{π} \frac{1}{t}*\cos \left(t\right)*\frac{-1}{t^2}dt$$
Next we have $$\cos \left(t\right) < 1 - \frac{t^2}{2}+\frac{t^4}{24}$$
Hence the given integral is smaller than
$$∫_{t=1}^\frac{1}{π} \frac{-1}{t^3}\left(1-\frac{t^2}{2}+\frac{t^4}{24}\right)dt=3.8811597.. $$
