Решебник по алгебре 11 класс Никольский Параграф 6. Первообразная и интеграл Задание 49

\[\boxed{\mathbf{49}.}\]

\[\textbf{а)}\ \int_{0}^{\pi}{\sin x\text{dx}};\]

\[F(x) = - \cos x;\]

\[\int_{0}^{\pi}{\sin x\text{dx}} = - \cos\pi + \cos 0 =\]

\[= - ( - 1) + 1 = 2.\]

\[\textbf{б)}\ \int_{- \frac{\pi}{2}}^{\frac{\pi}{2}}{\sin x\text{dx}};\]

\[F(x) = - \cos x;\]

\[\int_{- \frac{\pi}{2}}^{\frac{\pi}{2}}{\sin x\text{dx}} =\]

\[= - \cos\frac{\pi}{2} + \cos\left( - \frac{\pi}{2} \right) =\]

\[= - 0 + 0 = 0.\]

\[\textbf{в)}\ \int_{0}^{2\pi}{\sin x\text{dx}};\]

\[F(x) = - \cos x;\]

\[\int_{0}^{2\pi}{\sin x\text{dx}} =\]

\[= - \cos{2\pi} + \cos 0 = - 1 + 1 =\]

\[= 0.\]