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How to evaluate $$\int_0^1\frac{\arctan x}{x\sqrt{1-x^2}}\,\mathrm dx\text{?}$$ The steps I can think of is integration by parts as $$\int_0^1\frac{\arctan x}{x\sqrt{1-x^2}}\,\mathrm dx=\int_0^1\frac{\arctan x}{x}\,\mathrm d(\arcsin x)$$ or integration by substitution using $x=\sin t$ as $$\int_0^1 \frac{\arctan x}{x\sqrt{1-x^2}}\,\mathrm dx = \int_0^{\frac{\pi}{2}}\frac{\arctan(\sin t)}{\sin t}\,\mathrm dt,$$ but both seems to make the problem more complicated.

Thanks a lot.

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  • $\begingroup$ Have you tried substituting $x=\tan t$? $\endgroup$
    – Badr B
    Commented Jun 2, 2018 at 3:19
  • $\begingroup$ @BadrB Thanks. I know that $1+tan^2 t=sec^2 t$, but I'm not sure how to handle $\sqrt{1-tan^2t}$? $\endgroup$
    – yuan
    Commented Jun 2, 2018 at 3:27
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    $\begingroup$ @yuan Are you sure it's a textbook problem? $\endgroup$
    – Frank W
    Commented Jun 2, 2018 at 3:48
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    $\begingroup$ By drawing a diagram, we can see that $\tan t = \frac{\sqrt{1-x^2}}{x}$. Try substituting that value of $\tan t$ into the radical. $\endgroup$
    – Badr B
    Commented Jun 2, 2018 at 3:59
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    $\begingroup$ @BadrB : ok, Let's try it: $$ \begin{align} \text{We have } & \int_0^1 \frac{\arctan x}{x\sqrt{1-x^2}}\,\mathrm dx. \\ \\ \text{Let } & t = \arctan x \\ & \tan t=x \\ & \sec^2 t \, dt = dx \\ & 1-x^2 = \frac{\cos^2 t - \sin^2 t}{\cos^2 t} = \frac{\cos(2t)}{\cos^2 t} \\ \\ \text{The integral becomes } & \int_0^{\pi/2} \frac t {(\tan t)\frac{\sqrt{\cos (2t)}}{\cos t}} \cdot \frac{\mathrm dt}{\cos^2 t} = \int_0^{\pi/2} \frac t {(\sin t) \sqrt{\cos(2t)}} \, \mathrm dt \end{align} $$ Whither hence? $\qquad$ $\endgroup$ Commented Jun 2, 2018 at 5:12

2 Answers 2

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This can be solved using contour integration, which is overkill if this is truly a textbook problem, or it can be solved using the power of double integration (or nested integration, or whatever people call it).

Hopefully, you are aware that$$\frac {\arctan x}x=\int\limits_0^1dy\,\frac 1{1+x^2y^2}$$

If not, it's a good practice problem for you to try! Next, replace the $\arctan(\cdot)$ fraction with the above identity and switch the order of integration

$$\begin{align*}I & =\int\limits_0^1dy\,\int\limits_0^1 dx\,\frac 1{(1+x^2y^2)\sqrt{1-x^2}}\end{align*}$$

Now make the substitution $x\mapsto\sin x$ to get

$$\begin{align*}I & =\int\limits_0^1 dy\,\int\limits_0^{\pi/2}dx\,\frac 1{\cos^2 x+\sin^2 x(1+y^2)}\\ & =\int\limits_0^1dy\,\int\limits_0^{\pi/2}dx\,\frac {\sec^2x}{1+\tan^2x(1+y^2)}\\ & =\int\limits_0^1dy\,\int\limits_0^{\infty}dx\,\frac 1{1+x^2(1+y^2)}\end{align*}$$

Treating $1+y^2$ as a constant inside the second integral, we can pull a factor out so that the inner integral becomes a simple $\arctan(\cdot)$ question that we know how to evaluate!

$$\begin{align*}I & =\int\limits_0^1 dy\,\frac {\sqrt{1+y^2}}{1+y^2}\arctan\left(x\sqrt{1+y^2}\right)\,\Biggr\rvert_0^{\infty}\\ & =\frac {\pi}2\int\limits_0^1 dy\,\frac 1{\sqrt{1+y^2}}\\ & =\frac {\pi}2\operatorname{arcsinh} 1\end{align*}$$

Or, in a much more friendly notation free of hyperbolic functions, $I$ is also equal as

$$\int\limits_0^1 dx\,\frac {\arctan x}{x\sqrt{1-x^2}}\color{blue}{=\frac {\pi}2\log(1+\sqrt{2})}$$

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    $\begingroup$ Cool! Thank you very much for the great answer! It is really overkill, and takes me a while to go through the solution. $\endgroup$
    – yuan
    Commented Jun 2, 2018 at 5:07
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Let $\displaystyle I(a)=\int^{1}_{0}\frac{\tan^{-1}(ax)}{x\sqrt{1-x^2}}dx$

Now $\displaystyle I'(a)=\int^{1}_{0}\frac{1}{(1+a^2x^2)\sqrt{1-x^2}}dx$

Put $x=\tan t$. Then $dx=dt$ and changing limits

So $$I'(a)=\int^{\frac{\pi}{2}}_{0}\frac{1}{1+a^2\sin^2 t}dt=\frac{1}{1+a^2}\int^{\frac{\pi}{2}}_{0}\frac{\sec^2 t}{k^2+\tan^2 t}dt$$

Where $\displaystyle k^2=\frac{1}{1+a^2}$

So $\displaystyle I'(a)=\frac{1}{(1+a^2)\cdot k}\tan^{-1}\bigg(\frac{t}{k}\bigg)\bigg|^{\infty}_{0}=\frac{\pi}{2}\cdot \frac{1}{\sqrt{1+a^2}}$

So $$I(a)=\frac{\pi}{2}\ln\bigg|a+\sqrt{1+a^2}\bigg|$$

put $a=1$. Then $$I(1)=\frac{\pi}{2}\ln\bigg|1+\sqrt{2}\bigg|.$$

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