Вычисли определённый интеграл: 1) ∫x dx

Конечно, давай вычислим определённые интегралы! Вот решение для каждого из них: 1) $\int_{3}^{7} x dx = \frac{x^2}{2} \Big|_{3}^{7} = \frac{7^2}{2} - \frac{3^2}{2} = \frac{49}{2} - \frac{9}{2} = \frac{40}{2} = 20$ 2) $\int_{0}^{8} 2 dx = 2x \Big|_{0}^{8} = 2(8) - 2(0) = 16 - 0 = 16$ 3) $\int_{-5}^{5} x^2 dx = \frac{x^3}{3} \Big|_{-5}^{5} = \frac{5^3}{3} - \frac{(-5)^3}{3} = \frac{125}{3} - \frac{-125}{3} = \frac{125}{3} + \frac{125}{3} = \frac{250}{3}$ 4) $\int_{-2}^{2} x^4 dx = \frac{x^5}{5} \Big|_{-2}^{2} = \frac{2^5}{5} - \frac{(-2)^5}{5} = \frac{32}{5} - \frac{-32}{5} = \frac{32}{5} + \frac{32}{5} = \frac{64}{5}$ 5) $\int_{0}^{\pi} \sin(x) dx = -\cos(x) \Big|_{0}^{\pi} = -\cos(\pi) - (-\cos(0)) = -(-1) - (-1) = 1 + 1 = 2$ 6) $\int_{\frac{\pi}{4}}^{\frac{\pi}{3}} \frac{dx}{\cos^2(x)} = \tan(x) \Big|_{\frac{\pi}{4}}^{\frac{\pi}{3}} = \tan(\frac{\pi}{3}) - \tan(\frac{\pi}{4}) = \sqrt{3} - 1$ 7) $\int_{16}^{100} \frac{dx}{\sqrt{x}} = 2\sqrt{x} \Big|_{16}^{100} = 2\sqrt{100} - 2\sqrt{16} = 2(10) - 2(4) = 20 - 8 = 12$ 8) $\int_{1}^{e^3} \frac{dx}{x} = \ln(x) \Big|_{1}^{e^3} = \ln(e^3) - \ln(1) = 3 - 0 = 3$ 9) $\int_{1}^{10} \frac{dx}{x^2} = -\frac{1}{x} \Big|_{1}^{10} = -\frac{1}{10} - (-\frac{1}{1}) = -\frac{1}{10} + 1 = \frac{9}{10}$ 10) $\int_{-2}^{3} 3^x dx = \frac{3^x}{\ln(3)} \Big|_{-2}^{3} = \frac{3^3}{\ln(3)} - \frac{3^{-2}}{\ln(3)} = \frac{27}{\ln(3)} - \frac{1}{9\ln(3)} = \frac{243 - 1}{9\ln(3)} = \frac{242}{9\ln(3)}$ 11) $\int_{1}^{9} \sqrt{x} dx = \int_{1}^{9} x^{\frac{1}{2}} dx = \frac{x^{\frac{3}{2}}}{\frac{3}{2}} \Big|_{1}^{9} = \frac{2}{3} x^{\frac{3}{2}} \Big|_{1}^{9} = \frac{2}{3} (9^{\frac{3}{2}}) - \frac{2}{3} (1^{\frac{3}{2}}) = \frac{2}{3} (27) - \frac{2}{3} (1) = 18 - \frac{2}{3} = \frac{54 - 2}{3} = \frac{52}{3}$ 12) $\int_{-2}^{2} (2x + 4) dx = x^2 + 4x \Big|_{-2}^{2} = (2^2 + 4(2)) - ((-2)^2 + 4(-2)) = (4 + 8) - (4 - 8) = 12 - (-4) = 16$ 13) $\int_{0}^{2} (3x^2 - x) dx = x^3 - \frac{x^2}{2} \Big|_{0}^{2} = (2^3 - \frac{2^2}{2}) - (0^3 - \frac{0^2}{2}) = (8 - 2) - (0 - 0) = 6 - 0 = 6$ 14) $\int_{0}^{\frac{\pi}{4}} (4\sin(x) + 2\cos(x)) dx = -4\cos(x) + 2\sin(x) \Big|_{0}^{\frac{\pi}{4}} = (-4\cos(\frac{\pi}{4}) + 2\sin(\frac{\pi}{4})) - (-4\cos(0) + 2\sin(0)) = (-4 \cdot \frac{\sqrt{2}}{2} + 2 \cdot \frac{\sqrt{2}}{2}) - (-4 \cdot 1 + 2 \cdot 0) = -2\sqrt{2} + \sqrt{2} + 4 = 4 - \sqrt{2}$ Надеюсь, это поможет тебе в учёбе!