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

\[\boxed{\mathbf{12}.}\]

\[\textbf{а)}\ \int_{}^{}xdx = \int_{}^{}x^{1}dx =\]

\[= \int_{}^{}x^{2 - 1}dx = \frac{x^{2}}{2} + C.\]

\[\textbf{б)}\ \int_{}^{}x^{2}dx = \int_{}^{}x^{3 - 1}dx =\]

\[= \frac{x^{3}}{3} + C.\]

\[\textbf{в)}\ \int_{}^{}x^{3}dx = \int_{}^{}x^{4 - 1}dx =\]

\[= \frac{x^{4}}{4} + C.\]

\[\textbf{г)}\ \int_{}^{}{\sin x}dx = - \frac{\cos x}{1} + C =\]

\[= - \cos x + C.\]

\[\textbf{д)}\ \int_{}^{}{\cos x}dx = \frac{\sin x}{1} + C =\]

\[= \sin x + C.\]

\[\textbf{е)}\ \int_{}^{}\frac{\text{dx}}{\text{co}s^{2}x} = tg\ x + C.\]

\[\textbf{ж)}\ \int_{}^{}\frac{\text{dx}}{\text{si}n^{2}x} = - ctg\ x + C.\]

\[\textbf{з)}\ \int_{}^{}e^{x}dx = e^{x} + C.\]

\[\textbf{и)}\ \int_{}^{}8^{x}dx = \frac{8^{x}}{\ln 8} + C.\]

\[к)\ \int_{}^{}\frac{\text{dx}}{x} = \ln{|x|} + C.\]

\[л)\ \int_{}^{}x^{\frac{2}{3}}dx = \frac{x^{\frac{2}{3} + 1}}{\frac{2}{3} + 1} + C =\]

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

\[м)\ \int_{}^{}\sqrt{x}dx = \int_{}^{}x^{\frac{1}{2}}dx =\]

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

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