Вариант 1 3 ∫(x² + 4x - 1)dx; 0 3 ∫(3-1/x²)dx; 1 3 ∫(2/√(x+1) +3x²)dx; 0 π/2 1 x ∫ ---cos---dx. 0 2 2

\[\int (x^2 + 4x - 1) dx = \frac{x^3}{3} + 2x^2 - x + C\]

\[\left[\frac{x^3}{3} + 2x^2 - x\right]_{0}^{3} = \left(\frac{3^3}{3} + 2(3)^2 - 3\right) - \left(\frac{0^3}{3} + 2(0)^2 - 0\right) = \frac{27}{3} + 18 - 3 = 9 + 18 - 3 = 24\]

\[\int \left(3 - \frac{1}{x^2}\right) dx = 3x + \frac{1}{x} + C\]

\[\left[3x + \frac{1}{x}\right]_{1}^{3} = \left(3(3) + \frac{1}{3}\right) - \left(3(1) + \frac{1}{1}\right) = 9 + \frac{1}{3} - 3 - 1 = 5 + \frac{1}{3} = \frac{16}{3}\]

\[\int \left(\frac{2}{\sqrt{x+1}} + 3x^2\right) dx = 4\sqrt{x+1} + x^3 + C\]

\[\left[4\sqrt{x+1} + x^3\right]_{0}^{3} = \left(4\sqrt{3+1} + 3^3\right) - \left(4\sqrt{0+1} + 0^3\right) = 4\sqrt{4} + 27 - 4\sqrt{1} - 0 = 4(2) + 27 - 4 = 8 + 27 - 4 = 31\]

\[\int \frac{1}{2} \cos\left(\frac{x}{2}\right) dx = \sin\left(\frac{x}{2}\right) + C\]

\[\left[\sin\left(\frac{x}{2}\right)\right]_{0}^{\frac{\pi}{2}} = \sin\left(\frac{\pi}{4}\right) - \sin(0) = \frac{\sqrt{2}}{2} - 0 = \frac{\sqrt{2}}{2}\]