Although I have known that $\displaystyle\int_0^\infty {{\sin x} \over x} \, dx = {\pi \over 2}$, I have no idea how to work out $\displaystyle\int_0^{ + \infty } {{\cos x} \over x} \, dx$. How can I?
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3 Answers
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Unfortunately, the integral $$ \int_0^\infty\frac{\cos x}{x}\,dx, $$ diverges.
To see this observe that $$ \int_0^{\pi/4}\frac{\cos x}{x}\,dx\ge \frac{\sqrt{2}}{2}\int_0^{\pi/4}\frac{1}{x}=\infty. $$
answered Oct 17, 2014 at 18:00
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$$\int \frac{\cos x}{x} \, dx = \operatorname{Ci}(x) + C,$$ where $\operatorname{Ci}$ is the Cosine Integral.
Your integral does not converge on the interval $[0,\infty)$, since $\operatorname{Ci}(0)=-\infty$ and $\lim_{a \to \infty}\operatorname{Ci}(a)=0$
answered Oct 17, 2014 at 17:56
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The integral does not converge
simply the result is$$\int_0^{\infty} \frac{\cos x}{x} \, dx = \infty$$
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