If the value of the integral

$\int_{0}^{1 / 2} \frac{\left(\left(x^{2}-1\right)+1\right)}{\left(1-x^{2}\right)^{3 / 2}} d x$

$\int_{0}^{1 / 2} \frac{d x}{\left(1-x^{2}\right)^{3 / 2}}-\int_{0}^{1 / 2} \frac{d x}{\sqrt{1-x^{2}}}$

$\int_{0}^{1 / 2} \frac{x^{-3}}{\left(x^{-2}-1\right)^{3 / 2}} d x-\left(\sin ^{-1} x\right)_{0}^{1 / 2}$

Let $\mathrm{x}^{-2}-1=\mathrm{t}^{2} \Rightarrow \mathrm{x}^{-3} \mathrm{~d} \mathrm{x}=-\mathrm{tdt}$

$\int_{\infty}^{\sqrt{3}-t d t} \frac{\pi}{t^{3}}-\frac{\pi}{6}=\int_{\sqrt{3}}^{\infty} \frac{d t}{t^{2}}-\frac{\pi}{6}=\frac{1}{\sqrt{3}}-\frac{\pi}{6}=\frac{k}{6}$

$\mathrm{k}=2 \sqrt{3}-\pi$