$$\int_{}^{}{\frac{1}{x}\text{\ dx}} = ln|x|\int_{}^{}{a^{x}\text{\ dx}} = \frac{a^{x}}{\text{lna}}\int_{}^{}\text{tgx\ dx} = - ln|cosx|\int_{}^{}\text{ctgx\ dx} = ln|sinx|$$

$$\int_{}^{}{\frac{1}{\cos^{2}x}\text{\ dx}} = tgx\int_{}^{}{\frac{1}{\sin^{2}x}\text{\ dx}} = - ctgx\int_{}^{}{\frac{1}{x^{2} + a^{2}}\text{\ dx}} = \frac{1}{a}\text{arctg}\frac{x}{a}$$

$$\int_{}^{}{\frac{1}{\sqrt{a^{2} - x^{2}}}\text{\ dx}} = \arcsin\frac{x}{a}cos2x = \cos^{2}x - \sin^{2}x$$