\int \frac{dt}{t\sqrt{1-t^2}}=|t=\frac{1}{u}, \text{dt}=\left.-\frac{\text{du}}{u^2}\right|=-\int \frac{1}{\frac{1}{u}\sqrt{1-\frac{1}{u^2}}}\frac{\text{du}}{u^2}=-\int \frac{1}{u\sqrt{\frac{u^2-1}{u^2}}}\text{du}=-\int \frac{1}{\sqrt{u^2-1}}\text{du}