Find $\lim_{x\to 0} \int_x^{x+1} \sqrt {\arctan {t}}\space dt$

Question

$$\lim_{x\to 0} \int_x^{x+1} \sqrt {\arctan {t}}\space dt$$

I think it does not exist because we can't talk about limit from $x\to 0^-$, but what if we just look for $\lim_{x\to 0^+}$? I see there less and less area of the graph so it should be $0$, but how we show it?

Answer 1

Let $F(x)=\int_{0}^x \sqrt{\arctan t}dt$ Then, $$\lim_{x\to 0+}\int_{x}^{x+1}\sqrt{\arctan t}dt=\lim_{x\to 0+}(F(x+1)-F(x))\\=\lim_{x\to 0+}(F(x+1)-F(1))-\lim_{x\to 0+}(F(x)-F(0))+F(1)-F(0)\\=\int_{0}^1\sqrt{\arctan t}dt\\=\int_{0}^{\pi/4}\sqrt{\theta}\sec^2\theta d\theta$$ I am not sure if this can be further simplified.

Answer 2

Warning : I wonder myself about the validity of my answer.

Suppose that we expand the integrand as a Taylor series around $t=0$ to get for the very first terms $$\sqrt{\tan ^{-1}(t)}=\sqrt{t}-\frac{t^{5/2}}{6}+\frac{31 t^{9/2}}{360}+O\left(t^{11/2}\right)$$Let us do that up to $n$ terms; integrate between $x$ and $x+1$ and expand the result as a Taylor series around $x=0$. Depending on $n$, the following limits would be obtained $$L_{(1)}=\frac 23$$ $$L_{(2)}=\frac{13}{21}$$ $$L_{(3)}=\frac{8797}{13860}$$ $$L_{(4)}=\frac{782237}{1247400}$$ $$L_{(5)}=\frac{39914989}{63201600}$$ $$L_{(6)}=\frac{16448621317}{26165462400}$$ $$L_{(7)}=\frac{810914558349223}{1285770822336000}$$ $$L_{(8)}=\frac{50156186947762121}{79717790984832000}$$ $$L_{(9)}=\frac{478368186036207277667}{758913370175600640000}$$which converge, rather slowly, to the numerical value of $$\int_0^1 \sqrt{\tan ^{-1}(t)}\,dt$$ as Samrat Mukhopadhyay answered ($0.6298296269$, value for which inverse symbolic calculators did not find anything).

Answer 3

We have an integral of an inverse function: $$\begin{eqnarray*}\int_{0}^{1}\sqrt{\arctan t}\,dt &=&\sqrt{\frac{\pi}{4}}-\int_{0}^{\sqrt{\frac{\pi}{4}}}\tan(y^2)\,dy\\&=&\sqrt{\frac{\pi}{4}}-\sqrt{\frac{4}{\pi}}\sum_{k\geq 0}\frac{(4^k-1)(-1)^{k-1}B_{2k}}{(4k-1)(2k)!}\left(\frac{\pi^2}{4}\right)^k\end{eqnarray*}$$ where we exploited the Taylor series of the tangent function, written in terms of Bernoulli numbers.