TITLE: Evaluating the integral: $\int_{0}^{\infty} \frac{|2-2\cos(x)-x\sin(x)|}{x^4}~dx$
QUESTION [6 upvotes]: I am interested in evaluating the following integral:
$$
\int_{0}^{\infty} \frac{|2-2\cos(x)-x\sin(x)|}{x^4}~dx
$$
Using Matlab, Numerically it seems that the integral is convergent, 
but I'm not sure about it. How can we prove that the integral is
convergent or not? 
Many Thanks in advance.

REPLY [2 votes]: let $I=\int_{0}^{\infty}\dfrac{2-2\cos{x}-x\sin{x}}{x^4}dx$
then using integration by parts we have
$$I=-\dfrac{1}{3}\dfrac{2-2\cos{x}-x\sin{x}}{x^3}|_{0}^{\infty}+\dfrac{1}{3}\int_{0}^{\infty}\dfrac{\sin{x}-x\cos{x}}{x^3}dx$$
then $$I=\dfrac{1}{3}\int_{0}^{\infty}\dfrac{\sin{x}-x\cos{x}}{x^3}dx$$
and application of integration by parts yields
$$I=-\dfrac{1}{6}\lim_{x\to 0^{+}}\dfrac{\sin{x}-x\cos{x}}{x^2}+\dfrac{1}{6}\int_{0}^{\infty}\dfrac{x\sin{x}}{x^2}dx=\dfrac{\pi}{12}$$
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