Основные правила вычисления интегралов

Пример \(x > 0\)

$$\int \frac{dx}{\sqrt{x^2-1}} = \begin{vmatrix} t = \arccos{\frac{1}{x}} \\ x = \frac{1}{\cos{t}} \\ dx = \frac{\sin{t}}{\cos^2{t}}dt \end{vmatrix} = \int \frac{\sin{t}dt}{\cos^2{t}\sqrt{\frac{1}{\cos^2{t}}-1}} = \int \frac{\sin{t}\cos{t}dt}{\cos^2{t}\sin{t}} = \int \frac{dt}{\cos{t}} = $$ $$\frac{1}{2} \ln \left | \frac{1 + \sin{t}}{1 - \sin{t}} \right | + C = \frac{1}{2} \ln \left | \frac{1 + \sqrt{1 - \cos^2{t}}}{1 - \sqrt{1 - \cos^2{t}}} \right | + C = \frac{1}{2} \ln \left | \frac{1 + \sqrt{1 - \frac{1}{x^2}}}{1 - \sqrt{1 - \frac{1}{x^2}}} \right | + C = $$ $$\frac{1}{2} \ln \left | \frac{x + \sqrt{x^2-1}}{x - \sqrt{x^2-1}} \right | + C = \frac{1}{2} \ln \left | \left ( x + \sqrt{x^2-1} \right ) ^2 \right | + C = \ln \left | x + \sqrt{x^2-1} \right | + C$$

Получили табличный интеграл: .