3 Вычислить: 1) [ 3x3dx; 1 2) ∫ dx; 2 x2 π 2 π 3) cos xdx; 4) sin 2xdx. 0 2

Вычислим интегралы:

1) $$\int_{1}^{2} 3x^3 dx$$

• $$\int_{1}^{2} 3x^3 dx = 3 \int_{1}^{2} x^3 dx = 3 \cdot \frac{x^4}{4} \Big|_{1}^{2} = \frac{3}{4} (2^4 - 1^4) = \frac{3}{4} (16 - 1) = \frac{3}{4} \cdot 15 = \frac{45}{4} = 11.25$$

Ответ: 11.25

2) $$\int_{2}^{4} \frac{dx}{x^2}$$

• $$\int_{2}^{4} \frac{dx}{x^2} = \int_{2}^{4} x^{-2} dx = \frac{x^{-1}}{-1} \Big|_{2}^{4} = -\frac{1}{x} \Big|_{2}^{4} = -\frac{1}{4} - (-\frac{1}{2}) = -\frac{1}{4} + \frac{1}{2} = \frac{-1 + 2}{4} = \frac{1}{4} = 0.25$$

Ответ: 0.25

3) $$\int_{0}^{\frac{\pi}{2}} \cos x dx$$

• $$\int_{0}^{\frac{\pi}{2}} \cos x dx = \sin x \Big|_{0}^{\frac{\pi}{2}} = \sin(\frac{\pi}{2}) - \sin(0) = 1 - 0 = 1$$

Ответ: 1

4) $$\int_{\frac{\pi}{2}}^{\pi} \sin 2x dx$$

• $$\int_{\frac{\pi}{2}}^{\pi} \sin 2x dx = -\frac{1}{2} \cos 2x \Big|_{\frac{\pi}{2}}^{\pi} = -\frac{1}{2} (\cos 2\pi - \cos \pi) = -\frac{1}{2} (1 - (-1)) = -\frac{1}{2} (2) = -1$$

Ответ: -1