Вычислить интеграл (1004-1011).

Привет! Давай решим представленные интегралы. Для решения используем формулу Ньютона-Лейбница: $\int_{a}^{b} f(x) dx = F(b) - F(a)$, где $F(x)$ — первообразная функции $f(x)$. ### Номер 1004 1) $\int_{0}^{1} x dx = \left. \frac{x^2}{2} \right|_{0}^{1} = \frac{1}{2} - 0 = 0,5$ 2) $\int_{0}^{3} x^2 dx = \left. \frac{x^3}{3} \right|_{0}^{3} = \frac{27}{3} - 0 = 9$ 3) $\int_{-1}^{2} 3x^2 dx = \left. x^3 \right|_{-1}^{2} = 8 - (-1) = 9$ 5) $\int_{2}^{3} \frac{1}{x^2} dx = \int_{2}^{3} x^{-2} dx = \left. -x^{-1} \right|_{2}^{3} = \left. -\frac{1}{x} \right|_{2}^{3} = -\frac{1}{3} - (-\frac{1}{2}) = \frac{1}{2} - \frac{1}{3} = \frac{1}{6}$ 6) $\int_{1}^{2} \frac{1}{x^3} dx = \int_{1}^{2} x^{-3} dx = \left. \frac{x^{-2}}{-2} \right|_{1}^{2} = \left. -\frac{1}{2x^2} \right|_{1}^{2} = -\frac{1}{8} - (-\frac{1}{2}) = \frac{1}{2} - \frac{1}{8} = \frac{3}{8} = 0,375$ 7) $\int_{1}^{4} \sqrt{x} dx = \int_{1}^{4} x^{1/2} dx = \left. \frac{x^{3/2}}{3/2} \right|_{1}^{4} = \left. \frac{2}{3} \sqrt{x^3} \right|_{1}^{4} = \frac{2}{3}(8 - 1) = \frac{14}{3} = 4\frac{2}{3}$ ### Номер 1005 1) $\int_{1}^{e} \frac{1}{x} dx = \left. \ln|x| \right|_{1}^{e} = \ln e - \ln 1 = 1 - 0 = 1$ 2) $\int_{0}^{\ln 2} e^x dx = \left. e^x \right|_{0}^{\ln 2} = e^{\ln 2} - e^0 = 2 - 1 = 1$ 4) $\int_{-2\pi}^{\pi} \sin x dx = \left. -\cos x \right|_{-2\pi}^{\pi} = -\cos(\pi) - (-\cos(-2\pi)) = -(-1) + 1 = 1 + 1 = 2$ 5) $\int_{-2\pi}^{\pi} \sin 2x dx = \left. -\frac{1}{2}\cos 2x \right|_{-2\pi}^{\pi} = -\frac{1}{2}\cos(2\pi) - (-\frac{1}{2}\cos(-4\pi)) = -\frac{1}{2}(1) + \frac{1}{2}(1) = 0$ ### Номер 1006 1) $\int_{-3}^{2} (2x-3) dx = \left. (x^2 - 3x) \right|_{-3}^{2} = (4 - 6) - (9 - (-9)) = -2 - 18 = -20$ 2) $\int_{-2}^{-1} (5-4x) dx = \left. (5x - 2x^2) \right|_{-2}^{-1} = (-5 - 2) - (-10 - 8) = -7 - (-18) = 11$ 4) $\int_{0}^{1} (x^2 + 1) dx = \left. (\frac{x^3}{3} + x) \right|_{0}^{1} = (\frac{1}{3} + 1) - 0 = 1\frac{1}{3} = \frac{4}{3}$ 5) $\int_{0}^{2} (3x^2 - 4x + 5) dx = \left. (x^3 - 2x^2 + 5x) \right|_{0}^{2} = (8 - 8 + 10) - 0 = 10$