Derivate prime ed integrali 2

∫(x⁸ + 4x⁻³ - 2cosx + 3x⁵)dx = ∫x⁸dx + ∫4x⁻³dx + ∫-2cosxdx + ∫3x⁵dx

= x⁹/9 + 4x⁻²/-2 - 2senx + 3x⁶/6

-4 - x⁴ ≥ 0 => x⁶ ≤ -4

∫(8x³ + 4x⁻³ + 2x⁻⁴ - cos(1/3x))dx =

= [2x⁴/4 + 4x⁻²/-2 + 2x⁻³/-3 - sen(1/3x)]₂

= x⁴/2 - 2x⁻² - 2/3 x⁻³ - 3/4 sen(1/3x)

∫ x³e^-xdx = x⁴g(x) - (x⁴/4)(-e^-x)

∫ e^3xdx = (3e^3x)/3 = e^3x

∫ e^4xdx = e^4x/4

∫ e^3xdx = e^x/g

∫ e^7xdx = e^-7x/-7

= -x³e^-x + 3 ∫ x³e^-xdx =

= -x³e^-x + 3 [x e^-x/ -1 - ∫ 2x e^-x/ -1 dx] =

= -x³e^-x + 3 [-x²e^-x + 2 ∫ x e^-xdx] =

= -x³e^-x - 3x²e^-x + 6 ∫ x e^-xdx =

= -x³e^-x - 3x²e^-x + 6 (-x e^-x - e^-x)

∫ x²e^-xdx = x²e^-x - 2∫ x e^-xdx =

∫ x e^-xdx = x e^-x/ -∫ 1 e^-x/ -2 dx =

= -x²e^-x + 2 ∫ x e^-xdx

= - x e^-x + e^-xdx = -x e^-x - e^-x

f(x) = 4x³ - 3x

X = ℝ

∫(4x³ - 3x) dx = x⁴ - ³/₂ x²

lim_x→+∞ (4x³ - 3x) = +∞

m = lim_x→+∞ (4x³ - 3x) / x = lim_x→+∞ (4x² - 3) = +∞

4x³ - 3x > 0

x (4x² - 3) > 0

4x² - 3 = 0 ⟹ x² = ³/₄ ⟹ x = ± ^√3/₂

x > 0

4x² - 3 > 0

x < -^√3/₂ or x > ^√3/₂