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

Hint:
$$\begin{align}
∫_1^π x \cos \left(\frac1{x}\right)dx&=∫_{1}^{1/π}\frac{\cos {u}}{u}\left(-\frac{du}{u^2}\right)\\
&=∫_{1/π}^{1}\frac{\cos {u}}{u^3}du\\
&\leq∫_{1/π}^{1}\frac{1-\frac12u^2+\frac{1}{24}u^4}{u^3}du
\end{align}$$
