The integral is $$\begin{align*} I= \int_{0}^{1} \frac{1}{\sqrt[]{ (1-x^{2})(1-k^{2}x^{2}) }} \ dx \end{align*}$$ with $0<k<1$ . $x=1$ is a singular point of the integrand. I want to show this integral is convergent.
My attempt was $$\begin{align*} I = \int_{0}^{1} \frac{1}{\sqrt[]{ (1+x)(1-x)(1+kx)(1-kx) }} \ dx \leq \int_{0}^{1} \frac{1}{\sqrt[]{ (1-x)(1-kx) }} \ dx \end{align*}$$ Since $1+x, 1+kx$ are greater than one
Then I don't know how to approach this. I tried to use $$\begin{align*} \sqrt[]{ ab } \geq \frac{2}{\frac{1}{a}+\frac{1}{b}} \end{align*}$$ but it doesn't work. I also tried the change of variable, but I failed.
Thus, without computing this integral, how should I show this integral is convergent? Any help is appreciated.
Note that $(1-k^2 x^2)^{-1/2}$ is bounded in view of $k<1$, so your integral is bounded by a constant times the integral of $(1-x^2)^{-1/2}$, which you can compute explicitly (its antiderivative is $\arcsin x$).