Domain of Integral $F(X)=\int_{0}^{x}\frac{1}{\ln(2-t^2)}dt$

Question

I'm trying to find the domain of this integral function: $$F(X)=\int_0^{x}\frac{1}{\ln({2-t^2})}dt$$

I know that the the integrand function's domain is $(-\sqrt2;-1) \cup (-1;1)\cup(1;+\sqrt2)$, so I have to discuss those points.

• In $x\to\pm1$, the function is equal to $\frac1{\epsilon}$, where the infinitesimal is the one of the logarithm, so the order (standard: x, with $x\to0$) is less than $1$, so the integral converges.
• In $x\to\pm\sqrt2$, the functions is equal to $\frac1{\infty}$, so the integral converges.

I don't really understand why I am wrong: the integral diverges in $-1$ and $+1$ (according to Desmos), so the domain isn't $(-\sqrt2;+\sqrt2)$, but $(-1;+1)$. Thanks so much for explanation.

Answer 1

The domain of the integrand is indeed what you write. We can restrict to $x>0$, because of symmetry.

In order to see what happens for $1<x<\sqrt{2}$, we need to see whether both $$ \int_0^1\frac{1}{\ln(2-t^2)}\,dt \qquad\text{and}\qquad \int_1^x\frac{1}{\ln(2-t^2)}\,dt $$ converge.

Now $\ln a\le a-1$, so $\ln(2-t^2)\le 1-t^2$ and therefore, over $[0,1)$, $$ \frac{1}{\ln(2-t^2)}\ge\frac{1}{1-t^2} $$ and $$ \int_0^1\frac{1}{1-t^2}\,dt $$ diverges.