Решебник по алгебре 11 класс Никольский Параграф 6. Первообразная и интеграл Задание 12

$$\boxed{{\displaystyle \mathbf{12}.}}$$

$$\mathbf{\text{а)}}{\int }_{}^{}xdx={\int }_{}^{}{x}^{1}dx=$$

$$={\int }_{}^{}{x}^{2-1}dx=\frac{x^2}{2}+C.$$

$$\mathbf{\text{б)}}{\int }_{}^{}{x}^{2}dx={\int }_{}^{}{x}^{3-1}dx=$$

$$=\frac{x^3}{3}+C.$$

$$\mathbf{\text{в)}}{\int }_{}^{}{x}^{3}dx={\int }_{}^{}{x}^{4-1}dx=$$

$$=\frac{x^4}{4}+C.$$

$$\mathbf{\text{г)}}{\int }_{}^{}\sin xdx=-\frac{\cos x}{1}+C=$$

$$=-\cos x+C.$$

$$\mathbf{\text{д)}}{\int }_{}^{}\cos xdx=\frac{\sin x}{1}+C=$$

$$=\sin x+C.$$

$$\mathbf{\text{е)}}{\int }_{}^{}\frac{\text{dx}}{\text{co}{s}^{2}x}=tgx+C.$$

$$\mathbf{\text{ж)}}{\int }_{}^{}\frac{\text{dx}}{\text{si}{n}^{2}x}=-ctgx+C.$$

$$\mathbf{\text{з)}}{\int }_{}^{}{e}^{x}dx=e^x+C.$$

$$\mathbf{\text{и)}}{\int }_{}^{}{8}^{x}dx=\frac{8^x}{\ln 8}+C.$$

$$к){\int }_{}^{}\frac{\text{dx}}{x}=\ln |x|+C.$$

$$л){\int }_{}^{}{x}^{\frac{2}{3}}dx=\frac{x^{\frac{2}{3}+1}}{\frac{2}{3}+1}+C=$$

$$=\frac{x^{\frac{5}{3}}}{\frac{5}{3}}+C=\frac{3}{5}{x}^{\frac{5}{3}}+C.$$

$$м){\int }_{}^{}\sqrt{x}dx={\int }_{}^{}{x}^{\frac{1}{2}}dx=$$

$$=\frac{x^{\frac{1}{2}+1}}{\frac{1}{2}+1}+C=\frac{x^{\frac{3}{2}}}{\frac{3}{2}}+C=$$

$$=\frac{2}{3}{x}^{\frac{3}{2}}+C.$$