Таблица первообразных

$$\int x^{n} \, dx = \dfrac{x^{n+1}}{n+1} + C, ~~~ n\neq 1$$

$$\int \dfrac{1}{x} \, dx = \ln|x| + C$$

$$\int a^{x} \, dx = \dfrac{a^{x}}{\ln a} + C$$

$$\int e^{x} \, dx = e^{x} + C$$

$$\int \cos x \, dx = \sin x + C$$

$$\int \sin x \, dx = -\cos x + C$$

$$\int \dfrac{1}{\cos ^{2}x} \, dx = \mathrm{tg} x + C$$

$$\int \dfrac{1}{\sin ^{2}x} \, dx = -\mathrm{ctg} x + C$$

$$\int \dfrac{1}{\sqrt{1-x^{2}}} \, dx = \arcsin x + C$$

$$\int \dfrac{1}{1+x^{2}} \, dx = \mathrm{arctg} x + C$$

$$\int \dfrac{1}{\sqrt{x^{2} \pm 1}} \, dx = \ln|x + \sqrt{x^{2} \pm 1}| + C$$

$$\int \dfrac{1}{\sqrt{x^{2} \pm 1}} \, dx = \begin{cases} \mathrm{arsh} ~x + C, & + \\ \mathrm{arch} ~x + C, & - \end{cases}$$

$$\int \dfrac{1}{x^{2} - 1} \, dx = \dfrac{1}{2} \ln\left|\dfrac{x-1}{x+1}\right| + C$$

$$\int \dfrac{1}{x^{2} - 1} \, dx = \begin{cases} -\mathrm{arth} ~x + C_{1}, & |x| < 1 \\ -\mathrm{arcth} ~x + C_{2}, & |x| > 1 \end{cases}$$