# Proof of $\int_0^\infty \frac{\sin x}{\sqrt{x}}dx=\sqrt{\pi/2}$

Numerically it seems to be true that

$$\int_0^\infty \frac{\sin x}{\sqrt{x}}dx=\sqrt{\frac{\pi}{2}}.$$

Any ideas how to prove this?

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Substitute $u=\sqrt{x}$ to see that this is related to Fresnel's integral... –  draks ... Jul 17 '12 at 15:24

Using contour integration, we get \begin{align} \int_0^\infty\frac{e^{ix}}{\sqrt{x}}\,\mathrm{d}x &=\sqrt{i\,}\int_0^\infty\frac{e^{-x}}{\sqrt{x}}\,\mathrm{d}x\\ &=\frac{1+i}{\sqrt{2}}\Gamma\left(\frac12\right)\\ &=(1+i)\sqrt{\frac\pi2} \end{align} Therefore, $$\int_0^\infty\frac{\cos(x)}{\sqrt{x}}\,\mathrm{d}x=\int_0^\infty\frac{\sin(x)}{\sqrt{x}}\,\mathrm{d}x=\sqrt{\frac\pi2}$$

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That is a Fresnel integral.

Make the substitution $\sqrt{x}=u$. Then you get $dx=2udu$ from where

$$\int_0^\infty \sin(x) x^{-1/2} dx =2 \int_0^\infty \sin(u^2)du=2\sqrt{\frac{\pi}{8}}=\sqrt{\frac{\pi}{2}}$$

See this question for more details.

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Here is an another approach, which I show only a heuristic calculation:

\begin{align*} \int_{0}^{\infty} \frac{\sin x}{\sqrt{x}} \; dx &= \int_{0}^{\infty} \left( \frac{1}{\Gamma\left(\frac{1}{2}\right)} \int_{0}^{\infty} t^{-1/2} e^{-xt} \; dt \right) \sin x \; dx \\ &\stackrel{\ast}{=} \frac{1}{\Gamma\left(\frac{1}{2}\right)} \int_{0}^{\infty} t^{-1/2} \int_{0}^{\infty} e^{-xt} \sin x \; dx \; dt \\ &= \frac{1}{\Gamma\left(\frac{1}{2}\right)} \int_{0}^{\infty} \frac{t^{-1/2}}{1+t^2} \; dt \\ &= \frac{1}{\Gamma\left(\frac{1}{2}\right)} \int_{0}^{\infty} \frac{2du}{1+u^4}.\qquad(t = u^2) \end{align*}

Now it is not hard to show that

$$\int_{0}^{\infty} \frac{2du}{1+u^4} = \frac{\pi}{\sqrt{2}}.$$

Indeed, you may use the equality

\begin{align*} \frac{2 du}{1+u^4} &= \frac{2u^{-2} \; du}{u^2 + u^{-2}} = \frac{1 + u^{-2} \; du}{u^2 + u^{-2}} - \frac{1 - u^{-2} \; du}{u^2 + u^{-2}}\\ &= \frac{d\left(u - u^{-1}\right)}{\left(u - u^{-1}\right)^2 + 2} - \frac{d\left(u + u^{-1}\right)}{\left(u + u^{-1}\right)^2 - 2} \end{align*}

and hence deduce that

\begin{align*} \int_{0}^{\infty} \frac{2 du}{1+u^4} &= \int_{0}^{\infty} \frac{d\left(u - u^{-1}\right)}{\left(u - u^{-1}\right)^2 + 2} - \int_{0}^{\infty} \frac{d\left(u + u^{-1}\right)}{\left(u + u^{-1}\right)^2 - 2}\\ &= \int_{-\infty}^{\infty} \frac{dv}{v^2 + 2} - \color{blue}{\int_{\infty}^{\infty} \frac{dw}{w^2 - 2}} \qquad \begin{pmatrix}v = u - u^{-1} \\ w = u + u^{-1}\end{pmatrix}\\ &= \frac{\pi}{\sqrt{2}} + 0. \end{align*}

as claimed, where the blue-colored integration is taken along the curve starting from $+\infty$ to $2$, and then turning back to $+\infty$, which makes it cancel out. Therefore we obtain the desired result.

The problem in this calculation is that the starred equality is almost unable to be justified by any simple means.

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From the result

$$\int_0^\infty dx \frac{\sin x}{\sqrt x} = 2 \int_0^\infty dx \sin\left(x^2\right),$$

I'd use $\exp\left(-i x^2\right) = \cos \left(x^2\right) - i \sin \left(x^2\right)$ and

$$\int_0^\infty dx \ \exp\left(- a x^2\right) = \frac{1}{2} \sqrt{\frac{\pi}{a}}.$$

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Let's start out with the following relation: $$\int_0^{\infty} \frac{\sin x}{\sqrt{x}} e^{-a x} dx = \frac{2}{\sqrt{\pi}} \int_0^{\infty} \frac{1}{1+(a+x^2)^2} dx \tag1$$ Proof of the relation $(1)$

$$\int_0^{\infty} \frac{\sin x}{\sqrt{x}} e^{-a x} dx=$$ Notice that $\displaystyle \frac{1}{\sqrt x}= \frac{2}{\sqrt{\pi}} \int_{0}^{\infty} e^{-xt^2} dt$ and have that

$$\frac{2}{\sqrt{\pi}} \int_{0}^{\infty} \sin x e^{-ax}\left(\int_{0}^{\infty} e^{-xt^2} dt\right) dx=$$ $$\frac{2}{\sqrt{\pi}} \int_{0}^{\infty} \left(\int_{0}^{\infty}\sin x e^{-(a+t^2)x} dt\right) dx=$$ Change the integration order $$\frac{2}{\sqrt{\pi}} \int_{0}^{\infty} \left(\int_{0}^{\infty}\sin x e^{-(a+t^2)x} dx\right) dt=$$ Now let's recollect the formula $$\int e^{\alpha x} \sin (\beta x) \ dx = \frac{e^{\alpha x}(-\beta (\cos (\beta x) + \alpha \sin(\beta x)))}{{\alpha}^2+{\beta}^2}$$

Hence $$\int_{0}^{\infty}\sin x e^{-(a+t^2)x} dx=-\frac{e^{-(a+t^2)x}((a+t^2)\sin x + \cos x)}{1+(a+t^2)^2}\bigg|_{0}^{\infty}=\frac{1}{1+(a+t^2)^2}$$ Then $$\frac{2}{\sqrt{\pi}} \int_{0}^{\infty} \left(\int_{0}^{\infty}\sin x e^{-(a+t^2)x} dx\right) dt=\frac{2}{\sqrt{\pi}} \int_{0}^{\infty}\frac{1}{1+(a+t^2)^2} \ dt.$$ End of the relation $(1)$ proof.

Based upon the above relation we get that $$\int_0^{\infty} \frac{\sin x}{\sqrt{x}} dx=$$ $$\lim_{a\to0+} \int_0^{\infty} \frac{\sin x}{\sqrt{x}} e^{-a x} dx =$$ $$\lim_{a\to0+} \frac{2}{\sqrt{\pi}} \int_0^{\infty} \frac{1}{1+(a+x^2)^2} dx=$$ $$\frac{2}{\sqrt{\pi}} \int_0^{\infty} \frac{1}{1+x^4} dx \tag2$$ For the last integral we may change the variable and everything gets reduced to computing beta function

Change the variable $$x=\left(\frac{t}{1-t}\right)^{\frac{1}{4}}$$ Then $$\int_0^\infty \frac{1}{1+x^4} \ dx = \int_0^1 \frac{1}{4} (1-t)^{\frac{3}{4}-1} t^{\frac{1}{4}-1} \mathrm{d} t = \frac{1}{4} \operatorname{B}\left(\frac{1}{4}, \frac{3}{4}\right) = \frac{1}{4} \sqrt{2} \pi \tag3$$

Finally, from $(2)$ and $(3)$ we obtain the desired result

$$\int_0^\infty \frac{\sin x}{\sqrt{x}}dx=\sqrt{\frac{\pi}{2}}.$$ Q.E.D.

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Is it obvious what the result of the last indefinite integral is? –  Pedro Tamaroff Jul 17 '12 at 19:15
Still, don't write "obviously" when you mean to say that somebody already did the work for you; it's better to link to the answer you refer to instead. –  Ｊ. Ｍ. Jul 18 '12 at 11:21
@J. M.: agree. My answer was updated. –  Chris's sis Jul 18 '12 at 11:33