שינויים

<math>\int \frac{dx}{x\sqrt{1-x^{2}}}=\begin{Bmatrix}x=cosu\cos(u)\\ dx=sinudu\sin(u)du\end{Bmatrix}=\int \fracdfrac{sinu\sin(u)}{cosu\cos(u)\sqrt{1-\cos^{2}(u)}}du=\int \frac{du}{cosu\cos(u)}=</math>

<math>\begin{align}\int \frac{du}{cosu\cos(u)}&=\int \frac{2}{1+t^{2}}\cdot \frac{1+t^{2}}{1-t^{2}}dt=\int \frac{2dt}{(1+t)(1-t)}=\int\frac{dt}{1-t}+\frac{dt}{1+t}\\&=\ln\big(|1+t|\big)-\ln\big(|1-t|\big)+cC=\ln\left(\left|\frac{1+t}{1-t}\right|\right)+cC\end{align}</math>