# Evaluate $\int_0^1 x\arcsin\left(\sin\left(\frac 1x\right)\right) dx$

Evaluate $$\int_0^1 x\arcsin\left(\sin\left(\frac 1x\right)\right) dx=\int_1^{\infty} \frac {\arcsin(\sin x)}{x^3}dx$$

I tried to use Feynman's trick in the second one as

$$I(a)=\int_1^{\infty} \frac {\arcsin(\sin ax)}{x^3}dx$$

We have $$I(0)=0$$

Now differentiating both sides leads me to an absurd integral as $$I'(a)=\int_1^{\infty} \frac {\cos ax }{x^2\vert \cos ax \vert}dx$$

I am now unable proceed further.

Also If I keep in mind that I need value of integral at $$a=1$$ and so break the integral in intervals as $$\left(1,\frac {\pi}{2}\right) ;\left(\frac {\pi}{2},\frac {3\pi}{2}\right) ; \left(\frac {3\pi}{2},\frac {5\pi}{2}\right)$$ and so on then this gives me the value of integral as 0. But I suspect it isn't correct because one online software suggests it's value to be $$\frac {1}{2}$$

Moreover the signum function is always reminding me of the Grandi's series which also in some definition equals $$1/2$$

Any help would be greatly appreciated.

• Am I missing something here. Shouldn't $\operatorname{arcsin}(\sin(1/x)) = 1/x$
– user150203
Feb 22 '19 at 5:33
• @DavidG only for $x\in \left(\frac {2}{\pi}, 1\right)$ Feb 22 '19 at 5:41
• Yes of course. Thanks.
– user150203
Feb 22 '19 at 5:52

Write $$I_n = \left[\left(n-\frac{1}{2}\right)\pi, \left(n+\frac{1}{2}\right)\pi\right]$$ and notice that the behavior of $$\arcsin\sin x$$ on $$I_n$$ depends on the value of $$n$$. In particular, $$\arcsin\sin x$$ is linear on each interval $$I_n$$ with the slope $$(-1)^n$$. So, we can compute the integral as

\begin{align*} \int_{1}^{\infty} \frac{\arcsin\sin x}{x^3} \, \mathrm{d}x &= \left[ - \frac{\arcsin\sin x}{2x^2} \right]_{1}^{\infty} + \int_{1}^{\infty} \frac{(\arcsin\sin x)'}{2x^2} \, \mathrm{d}x \\ &= \frac{1}{2} + \int_{1}^{\infty} \frac{1}{2x^2} \left( \sum_{n=0}^{\infty} (-1)^n \mathbf{1}_{I_n}(x) \right) \, \mathrm{d}x \\ &= \frac{1}{2} + \sum_{n=0}^{\infty} (-1)^n \int_{I_n\cap[1,\infty)} \frac{1}{2x^2} \, \mathrm{d}x \\ &= \frac{1}{2} + \left[ - \frac{1}{2x} \right]_{1}^{\frac{\pi}{2}} + \sum_{n=1}^{\infty} (-1)^n \left[ - \frac{1}{2x} \right]_{\left(n-1/2\right)\pi}^{\left(n+1/2\right)\pi} \\ &= 1 - \sum_{n=1}^{\infty} \frac{(-1)^{n-1}}{\left( n - \frac{1}{2} \right) \pi} \\ &= 1 - \frac{1}{2} = \frac{1}{2}. \end{align*}

Or, more concisely,

\begin{align*} \int_{1}^{\infty} \frac{\arcsin\sin x}{x^3} \, \mathrm{d}x &= \left[ - \frac{\arcsin\sin x}{2x^2} \right]_{1}^{\infty} + \int_{1}^{\infty} \frac{(\arcsin\sin x)'}{2x^2} \, \mathrm{d}x \\ &= \frac{1}{2} + \left[ -\frac{(\arcsin\sin x)'}{2x} \right]_{1}^{\infty} + \int_{1}^{\infty} \frac{(\arcsin\sin x)''}{2x} \, \mathrm{d}x \\ &= 1 + \int_{1}^{\infty} \left( \sum_{n=1}^{\infty} \frac{(-1)^n}{x} \delta\left(x - \frac{2n-1}{2}\pi \right) \right) \, \mathrm{d}x \\ &= 1 + \sum_{n=1}^{\infty} \frac{(-1)^n}{\left(\frac{2n-1}{2}\right)\pi} = 1 - \frac{1}{2} = \frac{1}{2}. \end{align*}

• @Darkrai, It is indicator function. It takes value $1$ if $x \in I_n$, and $0$ otherwise. Feb 20 '19 at 17:33

To evaluate $$I(1)=\int_0^1 I^\prime(a) da$$ we only need $$I^\prime(a)$$ for $$a\in [0,\,1]$$, which guarantees $$\frac{\pi}{(2a)}>1$$. As you've already noted, $$I^\prime(a)=\int_1^{\pi/(2a)}\frac{dx}{x^2}-\sum_{n\ge 0}\left(\int_{\pi/(2a)+2n\pi/a}^{3\pi/(2a)+2n\pi/a}\frac{dx}{x^2}-\int_{3\pi/(2a)+2n\pi/a}^{5\pi/(2a)+2n\pi/a}\frac{dx}{x^2}\right)\\=1-\frac{2a}{\pi}-\frac{2a}{\pi}\sum_{n\ge 0}\left(\frac{1}{1+4n}-\frac{2}{3+4n}+\frac{1}{5+4n}\right).$$We'll come back to that infinite sum over $$n$$ later, but for now denote it by $$\Sigma$$ so$$I(1)=\int_0^1\left(1-\frac{2a(1+\Sigma)}{\pi}\right)da=1-\frac{1+\Sigma}{\pi}.$$So now we need to evaluate $$\Sigma$$; it's clearly $$1-2\left(\frac{1}{3}-\frac{1}{5}-\frac{1}{7}+\cdots\right)=1-2\left(1-\frac{\pi}{4}\right)=\frac{\pi}{2}-1,$$so$$I(1)=\tfrac{1}{2}.$$

I thought it might be instructive to present a way forward that does not use integration by parts, but rather evaluates the integral directly after enforcing the substitution $$x\mapsto 1/x$$. To that end, we proceed.

Note that we have

\begin{align} \int_1^\infty \frac{\arcsin(\sin(x))}{x^3}\,dx&=\int_1^{\pi/2}\frac{\arcsin(\sin(x))}{x^3}\,dx+\sum_{n=1}^\infty \int_{-\pi/2+n\pi}^{\pi/2+n\pi}\frac{\arcsin(\sin(x))}{x^3}\,dx\\\\ &=\int_1^{\pi/2}\frac{1}{x^2}\,dx+\sum_{n=1}^\infty (-1)^n\int_{-\pi/2}^{\pi/2}\frac{x}{(x+n\pi)^3}\,dx\\\\ &=\left(1-\frac2\pi\right)+\frac2\pi\sum_{n=1}^\infty (-1)^n\left(\frac{1}{2n-1}-\frac{1}{2n+1}+\frac{n}{(2n+1)^2}-\frac{n}{(2n-1)^2}\right)\\\\ &=\left(1-\frac2\pi\right)+\frac1\pi\sum_{n=1}^\infty (-1)^n\left(\frac{1}{2n-1}-\frac{1}{2n+1}-\frac{1}{(2n+1)^2}-\frac{1}{(2n-1)^2}\right)\\\\ &=\left(1-\frac2\pi\right)+\frac1\pi\left(-\frac\pi4+1-\frac\pi4+1\right)\\\\ &=\frac12 \end{align}

as expected!