Evaluation of Integral $\int_0^{+\infty}\frac{\sin^2(\tan\,\!x)}{x^2}\mathrm{d}x$

Question

Evaluation of $$\int_0^{+\infty}\dfrac{\sin^2(\tan\,\!x)}{x^2}\mathrm{d}x$$

Evaluate $\int_0^{\infty} {\sin(\tan(x)) \over x}dx$

I try to give the result through the method in the link, but I don't know what to do next.

\begin{align*} I&\overset{\mathrm{def}}{=} \int_0^{+\infty}\frac{\sin^2(\tan\,\!x)}{x^2}\,\mathrm{d}x\\ &= \frac12 \int_{-\infty}^{+\infty}\frac{\sin^2(\tan\,\!x)}{x^2}\,\mathrm{d}x\\ &=\frac12 \left(\sum_{n=-\infty}^\infty \int_{(n-\frac12)\pi}^{(n+\frac12)\pi}\right)\frac{\sin^2(\tan\,\!x)}{x^2}\,\mathrm{d}x \end{align*}

Answer 1

According to Lobachevsky Integral: $$\begin{aligned} &\int_0^{\infty}\frac{\sin^2(\tan x)}{x^2}\mathrm{d}x\\ =&\int_0^{\infty}\frac{\sin^2(\tan x)}{\sin^2x}\cdot \frac{\sin^2 x}{x^2}\mathrm{d}x\\ =&\int_0^{\frac{\pi}{2}}\frac{\sin^2(\tan x)}{\sin^2 x}\mathrm{d}x\\ =&\int_0^{\frac{\pi}{2}}\frac{\sin^2(\tan x)}{\tan^2 x}\mathrm{d}(\tan x)\\ =&\int_0^{\infty}\frac{\sin^2 u}{u^2}\mathrm{d}u\\ =&\frac{\pi}{2} \end{aligned}$$

Answer 2

The approach in the question seems to be attempting to follow the approach in my answer to the cited question, so I will adapt that approach here.

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Real Manipulations $$ \begin{align} \int_0^\infty\frac{\sin^2(\tan(x))}{x^2}\,\mathrm{d}x &=\frac12\int_{-\infty}^\infty\frac{\sin^2(\tan(x))}{x^2}\,\mathrm{d}x\tag1\\ &=\frac12\sum_{k\in\mathbb{Z}}\int_{-\pi/2}^{\pi/2}\frac{\sin^2(\tan(x))}{(x+k\pi)^2}\,\mathrm{d}x\tag2\\ &=\frac12\int_{-\pi/2}^{\pi/2}\frac{\sin^2(\tan(x))}{\sin^2(x)}\,\mathrm{d}x\tag3\\ &=\frac12\int_{-\infty}^\infty\frac{\sin^2(x)}{x^2}\,\mathrm{d}x\tag4\\[6pt] &=\frac\pi2\tag5 \end{align} $$ Explanation:
$(1)$: use the evenness of the integrand
$(2)$: use the periodicity of $\tan(x)$
$(3)$: substitute $x\mapsto x/\pi$ in $(7)$ from this answer and
$\phantom{\text{(3):}}$ take a derivative to get $\csc^2(x)=\sum\limits_{k\in\mathbb{Z}}\frac1{(x+k\pi)^2}$
$(4)$: substitute $x\mapsto\tan^{-1}(x)$
$(5)$: apply this answer

Answer 3

You can apply integration by parts and you get

$$\int_0^{\infty}\frac{\sin^2(\tan x)}{x^2}dx=\left.\frac{-\sin^2(\tan x)}{x}\right|_0^\infty+\int_0^{\infty}\frac{\sin(2\,\tan x)}{x}\frac{1}{\cos^2(x)}dx=$$ $$=\int_0^{\infty}\frac{\sin(2\,\tan x)}{x}\frac{1}{\cos^2(x)}dx.$$

Now, you can follow the same procedure that in Evaluate $\int_0^{\infty} {\sin(\tan(x)) \over x}dx$ to arrive to

$$\frac12\int_{-\frac12\pi}^{\frac12\pi}\frac{\sin (2\tan x)}{\tan x}\frac{1}{\cos^2(x)} dx$$

that performing the change of variable $\tan x\to u$

$$\frac12\int_{-\frac12\pi}^{\frac12\pi}\frac{\sin (2\tan x)}{\tan x}\frac{1}{\cos^2(x)} dx=\int_{-\infty}^{\infty}\frac{\sin (2u)}{2u} du=\int_{0}^{\infty}\frac{\sin (u)}{u} du=\frac{\pi}{2}$$

Answer 4

A different approach:

Consider the function $$f(z) = \frac{e^{i \alpha \tan z}}{z^{2}+\beta^{2}} , \quad (\alpha, \beta >0), $$ which has simple poles at $z= \pm i \beta$ and essential singularities at the half-integer multiples of $\pi$.

The magnitude of $e^{i \alpha \tan z}$ is $$\exp \left(\frac{-\alpha \sinh\left(2 \Im (z)\right)}{\cos\left(2 \Re (z)\right)+ \cosh \left(2 \Im (z)\right)} \right). $$

In the upper half plane, this value never exceeds $1$, which includes near the essential singularities on the real axis.

So by integrating $f(z)$ around a contour consisting of the real axis (with vanishingly small indentations about the half-integer multiples of $\pi$) and the large semicircle above it, we can conclude that $$\operatorname{PV}\int_{-\infty}^{\infty} \frac{e^{i \alpha \tan x}}{x^{2}+ \beta^{2}} \, \mathrm dx = 2 \pi i \operatorname*{Res}_{z=i \beta} f(z) = \frac{\pi}{\beta } \, e^{- \alpha \tanh \beta}. $$

Equating the real parts on both sides of the above equation, we have $$\int_{-\infty}^{\infty} \frac{\cos(\alpha \tan x)}{x^{2}+\beta^{2}} \, \mathrm dx = \frac{\pi}{\beta } \, e^{- \alpha \tanh \beta}.$$

Therefore, $$ \begin{align} \int_{0}^{\infty} \frac{\sin^{2}(\alpha \tan x)}{x^{2}} \, \mathrm dx &= \frac{1}{2} \int_{-\infty}^{\infty} \frac{\sin^{2}(\alpha \tan x)}{x^{2}} \, \mathrm dx \\ &= \frac{1}{2} \lim_{\beta \to 0^{+}} \int_{-\infty}^{\infty} \frac{\sin^{2}(\alpha \tan x)}{x^{2}+\beta^{2}} \, \mathrm dx \\ &= \frac{1}{4} \lim_{\beta \to 0^{+}} \int_{-\infty}^{\infty} \frac{1- \cos (2 \alpha \tan x)}{x^{2}+\beta^{2}} \, \mathrm dx \\ &= \frac{\pi }{4} \lim_{\beta \to 0^{+}} \frac{1- e^{- 2 \alpha \tanh \beta}}{\beta} = \\ &= \frac{\pi}{4} \lim_{\beta \to 0^{+}} \frac{2 \alpha \operatorname{sech}^{2}(\beta) \, e^{-2 \alpha \tanh \beta}}{1} \\ &= \frac{\pi \alpha}{2}. \end{align}$$