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Does the generalised integral $$\int_{0}^{\infty}\frac{e^{\arctan(x)}-1}{x \sqrt x}\,dx$$

converge or diverge?

The first thing I do is divide it into two integrals

$\int_{0}^{A}\frac{e^{\arctan(x)}-1}{x \sqrt x}\,dx$ + $\int_{A}^{\infty}\frac{e^{\arctan(x)}-1}{x \sqrt x}\,dx$.

And then I would want to utilize something like $\frac{e^{x}-1}{x}+\frac{1}{\sqrt x}$, where $\frac{1}{\sqrt x}$ is my $g(x)$.

I think I want to use $\int_{0}^{A}\frac{dx}{\sqrt x}$ and then calculate $\frac{f(x)}{g(x)}$ and then $\int_{A}^{\infty}\frac{e^{\arctan(x)}-1}{x \sqrt x}\,dx$.

But at this point I am a bit stuck with the calculations.

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Splitting the integral in two ranges, each including only one of the problematic points is wise. We can easily show that both of these integrals are finite as follows:

Consider the tail of the integral extending to infinity. We know that $\arctan(x)<\pi/2$ and thus we immediately conclude that

$$\int_{A}^{\infty}\frac{e^{\arctan x}-1}{x\sqrt{x}}dx<\int_{A}^{\infty}\frac{e^{\pi/2}-1}{x\sqrt{x}}dx=2\frac{e^{\pi/2}-1}{A^{1/2}}<\infty$$

For the first part finding a suitable bound is a bit less intuitive, but it's not that hard to tell that we need to put an upper bound on the function $f(x)=(e^x-1)/x$ which is constant in the vicinity of the origin. We take the derivative of $f$

$$f'(x)=\frac{(x-1)e^x+1}{x^2}\equiv\frac{g(x)}{x^2}$$

We examine the sign of $g$. The derivative of this function is $g'(x)=xe^x$ for $x>0$ and therefore we conclude that $g$ is increasing and thus

$$g(x)> g(0)\Rightarrow f'(x)> 0 ~~\forall~~x>0 $$

We finally have concluded that $f$ is itself and increasing function, and thus we have established that an upper bound for it is given by it's value at the rightmost boundary. Since $f(x)<f(A)$ and $\arctan x<x$ we get the estimate

$$\int_{0}^{A}\frac{e^{\arctan x}-1}{x\sqrt{x}}dx<\int_{0}^{A}\frac{e^x-1}{x\sqrt{x}}dx<\frac{e^A-1}{A}\int_{0}^A\frac{dx}{\sqrt{x}}=2\frac{e^A-1}{\sqrt{A}}<\infty$$

and the integral converges to a finite value.

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    $\begingroup$ Thanks a lot, I understand everything now, had some problems with g I realised $\endgroup$ – PythonDaniel Sep 15 '20 at 22:47
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What you could also have done is to check the integrand at the bounds.

Cloase to $x=0$ $$\frac{e^{\arctan(x)}-1}{x \sqrt x}=\frac 1{\sqrt x}\left(1+\frac{x}{2}+O\left(x^2\right) \right)$$ so, no problem at the lower bound.

For large values of $x$ $$\frac{e^{\arctan(x)}-1}{x \sqrt x}=\frac 1{x\sqrt x}\left(\left(e^{\pi /2}-1\right)-\frac{e^{\pi /2}}{x}+O\left(\frac{1}{x^2}\right) \right)$$ and still no problem.

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