Решебник по алгебре и начала математического анализа 11 класс Алимов Задание 1037

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\[1)\ \int_{0}^{\frac{\pi}{4}}{\frac{1}{2}\cos\left( x + \frac{\pi}{4} \right)\ \text{dx}} =\]

\[= \left. \ \frac{1}{2}\sin\left( x + \frac{\pi}{4} \right) \right|_{0}^{\frac{\pi}{4}} =\]

\[= \frac{1}{2}\sin\left( \frac{\pi}{4} + \frac{\pi}{4} \right) - \frac{1}{2}\sin\left( 0 + \frac{\pi}{4} \right) =\]

\[= \frac{1}{2} \bullet \sin\frac{\pi}{2} - \frac{1}{2} \bullet \sin\frac{\pi}{4} =\]

\[= \frac{1}{2} \bullet 1 - \frac{1}{2} \bullet \frac{\sqrt{2}}{2} = \frac{1}{2} - \sqrt{2} =\]

\[= \frac{2 - \sqrt{2}}{4};\]

\[2)\ \int_{0}^{\frac{\pi}{3}}{\frac{1}{3}\sin\left( x - \frac{\pi}{3} \right)\text{\ dx}} =\]

\[= \left. \ - \frac{1}{3}\cos\left( x - \frac{\pi}{3} \right) \right|_{0}^{\frac{\pi}{3}} =\]

\[= - \frac{1}{3} \bullet \cos 0 + \frac{1}{3}\cos\frac{\pi}{3} =\]

\[= - \frac{1}{3} \bullet 1 + \frac{1}{3} \bullet \frac{1}{2} =\]

\[= - \frac{1}{3} + \frac{1}{6} = - \frac{2}{6} + \frac{1}{6} = - \frac{1}{6};\]

\[3)\ \int_{1}^{3}{3\sin(3x - 6)\text{\ dx}} =\]

\[= \left. \ 3 \bullet \left( - \frac{1}{3}\cos(3x - 6) \right) \right|_{1}^{3} =\]

\[= \left. \ - \cos(3x - 6) \right|_{1}^{3} =\]

\[= - \cos(3 \bullet 3 - 6) + \cos(3 - 6) =\]

\[= - \cos 3 + \cos 3 = 0;\]

\[4)\ \int_{0}^{3}{8\cos(4x - 12)\text{\ dx}} =\]

\[= \left. \ 8 \bullet \frac{1}{4}\sin(4x - 12) \right|_{0}^{3} =\]

\[= \left. \ 2\sin(4x - 12) \right|_{0}^{3} =\]

\[= 2 \bullet \sin 0 + 2\sin 12 = 2\sin 12.\]