Prove that:
\left( \int_\pi^\infty\frac{\cos x}{x}\ dx\right)^{2} < \frac{1}{{\pi}^{2}}
Solution
Integrate by parts:
\int \frac{\cos x}{x} dx = \frac{\sin x}{x} + \int \frac{\sin x}{x^2} dx
So
\int_\pi^\infty\frac{\cos x}{x}\ dx = \int_\pi^\infty \frac{\sin x}{x^2} dx
And
| \int_\pi^\infty\frac{\cos x}{x}\ dx| = |\int_\pi^\infty \frac{\sin x}{x^2} dx|
\leq \int_\pi^\infty \frac{| \sin x |}{x^2} dx
\leq \int_\pi^\infty \frac{1}{x^2} dx
= \frac{1}{\pi}
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