Решебник по алгебре 11 класс Никольский Параграф 6. Первообразная и интеграл Задание 15

\[\boxed{\mathbf{15}.}\]

\[\textbf{а)}\ \int_{}^{}{\left( 5\sin{2x} - 3\cos\frac{x}{2} \right)\text{dx}} =\]

\[= 5\int_{}^{}{\sin{2x}\text{dx}} - 3\int_{}^{}{\cos\frac{x}{2}\text{dx}} =\]

\[= - \frac{5}{2}\cos{2x} - 3 \cdot \frac{\sin\frac{x}{2}}{\frac{1}{2}} + C =\]

\[= - 2,5\cos{2x} - 6\sin\frac{x}{2} + C.\]

\[\textbf{б)}\ \int_{}^{}{\left( \frac{5}{x + 1} - e^{5x - 1} \right)\text{dx}} =\]

\[= 5\int_{}^{}{\left( \frac{1}{x + 1} \right)\text{dx}} + \int_{}^{}{e^{5x - 1}\text{dx}} =\]

\[= 5\ln{|x + 1| - \frac{1}{5}e^{5x - 1}} + C.\]

\[= \int_{}^{}{\sqrt{(x + 1)^{3}} - \sqrt[3]{(x - 3)^{2}}} =\]

\[= \int_{}^{}{\left( (x + 1)^{\frac{3}{2}} - (x - 3)^{\frac{2}{3}} \right)\text{dx}} = \ \]

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

\[= \frac{2}{5}(x + 1)^{\frac{5}{2}} - \frac{3}{5}(x - 1)^{\frac{5}{3}} + C.\]