Решебник по алгебре 11 класс Никольский Параграф 6. Первообразная и интеграл Задание 13

\[\boxed{\mathbf{13}.}\]

\[\textbf{а)}\ \int_{}^{}{\left( x + \sin x \right)\text{dx}} =\]

\[= \int_{}^{}\text{xdx} + \int_{}^{}{\sin x\text{dx}} =\]

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

\[\textbf{б)}\ \int_{}^{}{\left( x^{2} - \cos x \right)\text{dx}} =\]

\[= \int_{}^{}{x^{2}\text{dx}} - \int_{}^{}{\cos x\text{dx}} =\]

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

\[\textbf{в)}\ \int_{}^{}{\left( \sqrt{x} - \frac{1}{\text{co}s^{2}x} \right)\text{dx}} =\]

\[= \int_{}^{}{x^{\frac{1}{2}}\text{dx}} - \int_{}^{}\frac{\text{dx}}{\text{co}s^{2}x} =\]

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

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

\[\textbf{г)}\ \int_{}^{}{\left( x^{\frac{3}{4}} + \frac{1}{\text{si}n^{2}x} \right)\text{dx}} =\]

\[= \int_{}^{}{x^{\frac{3}{4}}\text{dx}} + \int_{}^{}\frac{\text{dx}}{\text{si}n^{2}x} =\]

\[= \frac{x^{\frac{3}{4} + 1}}{\frac{3}{4} + 1} - ctg\ x + C =\]

\[= \frac{x^{\frac{7}{4}}}{\frac{7}{4}} - ctg\ x + C =\]

\[= \frac{4x^{\frac{7}{4}}}{7} - ctg\ x + C.\]

\[\textbf{д)}\ \int_{}^{}{\left( e^{x} - \frac{1}{x} \right)\text{dx}} =\]

\[= \int_{}^{}{e^{x}\text{dx}} - \int_{}^{}\frac{\text{dx}}{x} =\]

\[= e^{x} - \ln|x| + C.\]

\[\textbf{е)}\ \int_{}^{}{\left( 6^{x} + \frac{1}{x} \right)\text{dx}} =\]

\[= \int_{}^{}\frac{6^{x}}{\ln 6} + \int_{}^{}\frac{\text{dx}}{x} =\]

\[= \frac{6^{x}}{\ln 6} + \ln{|x|} + C.\]