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

Решение:

a) ∫2x³ dx

• ∫2x³ dx = 2 * ∫x³ dx = 2 * (x⁴/4) + C = x⁴/2 + C

b) ∫₂⁴ 1/x² dx

• ∫₂⁴ 1/x² dx = ∫₂⁴ x⁻² dx = [-x⁻¹]₂⁴ = [-1/x]₂⁴ = -1/4 - (-1/2) = -1/4 + 1/2 = 1/4

c) ∫₁⁴ 1/x dx

• ∫₁⁴ 1/x dx = [ln|x|]₁⁴ = ln(4) - ln(1) = ln(4) - 0 = ln(4)

d) ∫₀^(π/2) sin 2x dx

• ∫₀^(π/2) sin 2x dx = [-1/2 * cos 2x]₀^(π/2) = -1/2 * cos(π) - (-1/2 * cos(0)) = -1/2 * (-1) + 1/2 * 1 = 1/2 + 1/2 = 1

e) ∫₁⁴ 1/√x dx

• ∫₁⁴ 1/√x dx = ∫₁⁴ x^(-1/2) dx = [2x^(1/2)]₁⁴ = 2√4 - 2√1 = 2 * 2 - 2 * 1 = 4 - 2 = 2

Ответы:

• a) x⁴/2 + C
• b) 1/4
• c) ln(4)
• d) 1
• e) 2