3 Вычислить: 1) ∫ 3x³ dx; 1 2 2) ∫ dx; 2 x² 3) ∫ cos xdx; 0 π 4) ∫ sin 2xdx. π 2 π 2

• Вычисляем интеграл 1:

\[\int_{1}^{2} 3x^3 dx = 3 \int_{1}^{2} x^3 dx = 3 \left[ \frac{x^4}{4} \right]_{1}^{2} = 3 \left( \frac{2^4}{4} - \frac{1^4}{4} \right) = 3 \left( \frac{16}{4} - \frac{1}{4} \right) = 3 \left( \frac{15}{4} \right) = \frac{45}{4}\] \[\int_{1}^{2} 3x^3 dx = \frac{45}{4} = 11.25\]

• Вычисляем интеграл 2:

\[\int_{2}^{4} \frac{dx}{x^2} = \int_{2}^{4} x^{-2} dx = \left[ -x^{-1} \right]_{2}^{4} = \left[ -\frac{1}{x} \right]_{2}^{4} = -\frac{1}{4} - \left( -\frac{1}{2} \right) = -\frac{1}{4} + \frac{1}{2} = \frac{1}{4}\] \[\int_{2}^{4} \frac{dx}{x^2} = \frac{1}{4} = 0.25\]

• Вычисляем интеграл 3:

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

• Вычисляем интеграл 4:

\[\int_{\frac{\pi}{2}}^{\pi} \sin 2x dx = \left[ -\frac{1}{2} \cos 2x \right]_{\frac{\pi}{2}}^{\pi} = -\frac{1}{2} \cos(2\pi) - \left( -\frac{1}{2} \cos(\pi) \right) = -\frac{1}{2}(1) + \frac{1}{2}(-1) = -\frac{1}{2} - \frac{1}{2} = -1\] \[\int_{\frac{\pi}{2}}^{\pi} \sin 2x dx = -1\]