Вычислите следующие интегралы: a) ∫₂⁴ 2x³ dx ; b) ∫₁⁴ 1/x² dx ; c) ∫₁² 1/x dx; d) ∫₀^π/4 sin 2x dx ; e) ∫₄⁹ 1/√x dx;

Решение:

a) \( \int_2^4 2x^3 dx \)

• \( \int 2x^3 dx = 2 \int x^3 dx = 2 \cdot \frac{x^4}{4} = \frac{x^4}{2} \)
• \( \frac{x^4}{2} \Big|_2^4 = \frac{4^4}{2} - \frac{2^4}{2} = \frac{256}{2} - \frac{16}{2} = 128 - 8 = 120 \)

b) \( \int_1^4 \frac{1}{x^2} dx \)

• \( \int \frac{1}{x^2} dx = \int x^{-2} dx = \frac{x^{-1}}{-1} = -\frac{1}{x} \)
• \( -\frac{1}{x} \Big|_1^4 = -\frac{1}{4} - (-\frac{1}{1}) = -\frac{1}{4} + 1 = \frac{3}{4} \)

c) \( \int_1^2 \frac{1}{x} dx \)

• \( \int \frac{1}{x} dx = ln|x| \)
• \( ln|x| \Big|_1^2 = ln(2) - ln(1) = ln(2) - 0 = ln(2) \)

d) \( \int_0^{\frac{\pi}{4}} sin(2x) dx \)

• \( \int sin(2x) dx = -\frac{1}{2} cos(2x) \)
• \( -\frac{1}{2} cos(2x) \Big|_0^{\frac{\pi}{4}} = -\frac{1}{2} cos(\frac{\pi}{2}) - (-\frac{1}{2} cos(0)) = -\frac{1}{2} \cdot 0 + \frac{1}{2} \cdot 1 = \frac{1}{2} \)

e) \( \int_4^9 \frac{1}{\sqrt{x}} dx \)

• \( \int \frac{1}{\sqrt{x}} dx = \int x^{-\frac{1}{2}} dx = \frac{x^{\frac{1}{2}}}{\frac{1}{2}} = 2\sqrt{x} \)
• \( 2\sqrt{x} \Big|_4^9 = 2\sqrt{9} - 2\sqrt{4} = 2 \cdot 3 - 2 \cdot 2 = 6 - 4 = 2 \)