5. Вычислить интеграл: 1) ∫ -1 x²dx; 2) ∫ 0 2xdx; 3) ∫ 2 4 1/x²dx; 4) ∫ π 2π sin xdx; 5) ∫ 1 3 1/√x dx

Решение:

1. \( \int_{-1}^{5} x^2 dx = \left[ \frac{x^3}{3} \right]_{-1}^{5} = \frac{5^3}{3} - \frac{(-1)^3}{3} = \frac{125}{3} - \frac{-1}{3} = \frac{125+1}{3} = \frac{126}{3} = 42 \)
2. \( \int_{0}^{2} 2x dx = \left[ x^2 \right]_{0}^{2} = 2^2 - 0^2 = 4 - 0 = 4 \)
3. \( \int_{2}^{4} \frac{1}{x^2} dx = \int_{2}^{4} x^{-2} dx = \left[ \frac{x^{-1}}{-1} \right]_{2}^{4} = \left[ -\frac{1}{x} \right]_{2}^{4} = (-\frac{1}{4}) - (-\frac{1}{2}) = -\frac{1}{4} + \frac{2}{4} = \frac{1}{4} \)
4. \( \int_{\pi}^{2\pi} \sin x dx = \left[ -\cos x \right]_{\pi}^{2\pi} = -\cos(2\pi) - (-\cos(\pi)) = -1 - (-(-1)) = -1 - 1 = -2 \)
5. \( \int_{1}^{3} \frac{1}{\sqrt{x}} dx = \int_{1}^{3} x^{-\frac{1}{2}} dx = \left[ \frac{x^{\frac{1}{2}}}{\frac{1}{2}} \right]_{1}^{3} = \left[ 2\sqrt{x} \right]_{1}^{3} = 2\sqrt{3} - 2\sqrt{1} = 2\sqrt{3} - 2 \)

Ответ: 1) 42; 2) 4; 3) \( \frac{1}{4} \); 4) -2; 5) \( 2\sqrt{3} - 2 \).