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МатематикаРешебник по алгебре и начала математического анализа 11 класс Колягин Задание 370
Задание 370
\[1)\ \int_{- 2}^{- \frac{3}{2}}{( - 2 - 3x)\text{dx}} =\]
\[= \left. \ \left( - 2x - \frac{3}{2}x^{2} \right) \right|_{- 2}^{- \frac{3}{2}} =\]
\[= \left( 3 - \frac{27}{8} \right) - (4 - 6) =\]
\[= 5 - 3\frac{3}{8} = 1\frac{5}{8}.\]
\[2)\ \int_{- 2}^{3}{(6 - 2x)\text{dx}} = \left. \ \left( 6x - x^{2} \right) \right|_{- 2}^{3} =\]
\[= \left( 6 \bullet 3 - 3^{2} \right) - \left( 6 \bullet ( - 2) - ( - 2)^{2} \right) =\]
\[= (18 - 9) - ( - 12 - 4) =\]
\[= 9 + 16 = 25.\]
\[3)\ \int_{0}^{4}{\left( 12 + x - x^{2} \right)\text{dx}} =\]
\[= \left. \ \left( 12x + \frac{x^{2}}{2} - \frac{x^{3}}{3} \right) \right|_{0}^{4} =\]
\[= 12 \bullet 4 + \frac{4^{2}}{2} - \frac{4^{3}}{3} =\]
\[= 48 + \frac{16}{2} - \frac{64}{3} =\]
\[= 56 - 21\frac{1}{3} = 34\frac{2}{3}.\]
\[4)\ \int_{- 2}^{0}{\left( x^{2} - 3x \right)\text{dx}} =\]
\[= \left. \ \left( \frac{x^{3}}{3} - \frac{3x^{2}}{2} \right) \right|_{- 2}^{0} =\]
\[= - \left( \frac{( - 2)^{3}}{3} - \frac{3}{2} \bullet ( - 2)^{2} \right) =\]
\[= - \left( - \frac{8}{3} - \frac{3}{2} \bullet 4 \right) =\]
\[= \frac{8}{3} + 6 = 2\frac{2}{3} + 6 = 8\frac{2}{3}.\]
\[5)\ \int_{2\pi}^{3\pi}{\sin x\text{dx}} = \left. \ - \cos x \right|_{2\pi}^{3\pi} =\]
\[= - \cos{3\pi} + \cos{2\pi} =\]
\[= - ( - 1) + 1 = 1 + 1 = 2.\]
\[6)\ \int_{- \frac{\pi}{2}}^{\frac{\pi}{2}}{\cos x\text{dx}} = \left. \ \sin x \right|_{- \frac{\pi}{2}}^{\frac{\pi}{2}} =\]
\[= \sin\frac{\pi}{2} - \sin\left( - \frac{\pi}{2} \right) =\]
\[= 1 - ( - 1) = 1 + 1 = 2.\]
\[7)\ \int_{- \frac{\pi}{4}}^{\frac{\pi}{2}}{\cos x\text{dx}} = \left. \ \sin x \right|_{- \frac{\pi}{4}}^{\frac{\pi}{2}} =\]
\[= \sin\frac{\pi}{2} - \sin\left( - \frac{\pi}{4} \right) =\]
\[= 1 - \left( - \frac{\sqrt{2}}{2} \right) = \frac{\sqrt{2}}{2} + 1.\]
\[8)\ \int_{\frac{\pi}{6}}^{\pi}{\sin x\text{dx}} = \left. \ - \cos x \right|_{\frac{\pi}{6}}^{\pi} =\]
\[= - \cos\pi + \cos\frac{\pi}{6} =\]
\[= - ( - 1) + \frac{\sqrt{3}}{2} = \frac{\sqrt{3}}{2} + 1.\]
