Indefinite integral convergence How can I prove that this integral converges?
$$
\int_0^1 \sqrt{\frac{1-kx^2}{1-x^2}} dx\quad\quad 0\leq k<1
$$
Edit: fixed typo dt -> dx

Call $f(x):=\sqrt{\frac{1-kx^2}{1-x^2}}$. Then $f$ is well-defined on the open unit interval and non-negative. Since $\lim_{x\to 0}f(x)=1$, we only have to show that $\int_{1/2}^1f(x)\mathrm dx$ is convergent. This follows from the convergence of $\int_{1/2}^1\frac 1{\sqrt{1-x}}\mathrm{d}x$. 
