Для функции у(х), найдите F(x) Вычислите определенный интеграл. 1) ∫₋₁² x³ dx = Б). ∫₂³ x² dx = В) ∫₀¹ (3x²+x) dx = 1) ∫₋₁¹ (2x+2-5x+3) dx = 2) ∫₂⁵ (4x+1) dx = 3) ∫₀³ (1/√x + 1/x³ + x) dx = 4) ∫₄⁵ (6x-10) dx = 5) ∫₀² (1/2x + x² + 6) dx = 6) ∫₋₁¹ (6x³+5x²-x+3) dx = 7) ∫₀¹ (x/3 + x/x² + 3/x) dx = 8) ∫₋₂³ (x + 5 + x/4) dx = 9) ∫₁² (5x⁴ - 6x² + x) dx = 10) ∫₋₂² (6x + 3) dx = 11) ∫₋₂¹ (x² - x) dx = 12) ∫₄⁶ (3 + x/2) dx = 13) ∫₁⁹ 6/√x dx = 14) ∫₋₂⁴ (8 + 2x - x²) dx = 15) ∫₋₂³ (4x³ - 3x² + 2x + 1) dx = 16) ∫₁² (x⁴ + 1) dx = 17) ∫₀.₅¹ dx/x³ = 18) ∫₁⁹ (x-1)/√x dx = 19) ∫₁⁸ ³√x² dx =

Краткое пояснение:

1)

• ∫₋₁² x³ dx = [x⁴/4]₋₁² = (2⁴/4) - ((-1)⁴/4) = 16/4 - 1/4 = 15/4 = 3.75

Б)

• ∫₂³ x² dx = [x³/3]₂³ = (3³/3) - (2³/3) = 27/3 - 8/3 = 19/3 ≈ 6.33

В)

• ∫₀¹ (3x²+x) dx = [x³ + x²/2]₀¹ = (1³ + 1²/2) - (0³ + 0²/2) = 1 + 1/2 = 3/2 = 1.5

1)

• ∫₋₁¹ (2x+2-5x+3) dx = ∫₋₁¹ (-3x + 5) dx = [-3x²/2 + 5x]₋₁¹ = (-3(1)²/2 + 5(1)) - (-3(-1)²/2 + 5(-1)) = (-3/2 + 5) - (-3/2 - 5) = -3/2 + 5 + 3/2 + 5 = 10

2)

• ∫₂⁵ (4x+1) dx = [2x² + x]₂⁵ = (2(5)² + 5) - (2(2)² + 2) = (50 + 5) - (8 + 2) = 55 - 10 = 45

3)

• ∫₀³ (1/√x + 1/x³ + x) dx = ∫₀³ (x⁻¹/₂ + x⁻³ + x) dx = [2x¹/₂ - 1/(2x²) + x²/2]₀³ = (2√(3) - 1/(2(3²)) + 3²/2) - (2√(0) - 1/(2(0²)) + 0²/2) = (2√(3) - 1/18 + 9/2) - (0 - ∞ + 0) = 2√3 - 1/18 + 9/2 + ∞ = ∞ (не существует, т.к. интеграл от 1/x³ в точке 0 не определен)

4)

• ∫₄⁵ (6x-10) dx = [3x² - 10x]₄⁵ = (3(5)² - 10(5)) - (3(4)² - 10(4)) = (75 - 50) - (48 - 40) = 25 - 8 = 17

5)

• ∫₀² (1/2x + x² + 6) dx = [x²/4 + x³/3 + 6x]₀² = ((2)²/4 + (2)³/3 + 6(2)) - ((0)²/4 + (0)³/3 + 6(0)) = (4/4 + 8/3 + 12) - (0 + 0 + 0) = 1 + 8/3 + 12 = 13 + 8/3 = 47/3 ≈ 15.67

6)

• ∫₋₁¹ (6x³+5x²-x+3) dx = [6x⁴/4 + 5x³/3 - x²/2 + 3x]₋₁¹ = (6(1)⁴/4 + 5(1)³/3 - (1)²/2 + 3(1)) - (6(-1)⁴/4 + 5(-1)³/3 - (-1)²/2 + 3(-1)) = (6/4 + 5/3 - 1/2 + 3) - (6/4 - 5/3 - 1/2 - 3) = 6/4 + 5/3 - 1/2 + 3 - 6/4 + 5/3 + 1/2 + 3 = 10/3 + 6 = 28/3 ≈ 9.33

7)

• ∫₀¹ (x/3 + x/x² + 3/x) dx = ∫₀¹ (x/3 + 1/x + 3/x) dx = ∫₀¹ (x/3 + 4/x) dx = [x²/6 + 4ln|x|]₀¹ = (1²/6 + 4ln|1|) - (0²/6 + 4ln|0|) = 1/6 + 4(0) - 0 - 4(-∞) = 1/6 + ∞ = ∞ (не существует, т.к. интеграл от 1/x в точке 0 не определен)

8)

• ∫₋₂³ (x + 5 + x/4) dx = [x²/2 + 5x + x²/8]₋₂³ = ((3)²/2 + 5(3) + (3)²/8) - ((-2)²/2 + 5(-2) + (-2)²/8) = (9/2 + 15 + 9/8) - (4/2 - 10 + 4/8) = (36/8 + 120/8 + 9/8) - (16/8 - 80/8 + 4/8) = 165/8 - (-60/8) = 165/8 + 60/8 = 225/8 = 28.125

9)

• ∫₁² (5x⁴ - 6x² + x) dx = [x⁵ - 2x³ + x²/2]₁² = ((2)⁵ - 2(2)³ + (2)²/2) - ((1)⁵ - 2(1)³ + (1)²/2) = (32 - 16 + 4/2) - (1 - 2 + 1/2) = (16 + 2) - (-1 + 1/2) = 18 - (-1/2) = 18 + 1/2 = 37/2 = 18.5

10)

• ∫₋₂² (6x + 3) dx = [3x² + 3x]₋₂² = (3(2)² + 3(2)) - (3(-2)² + 3(-2)) = (12 + 6) - (12 - 6) = 18 - 6 = 12

11)

• ∫₋₂¹ (x² - x) dx = [x³/3 - x²/2]₋₂¹ = ((1)³/3 - (1)²/2) - ((-2)³/3 - (-2)²/2) = (1/3 - 1/2) - (-8/3 - 4/2) = (2/6 - 3/6) - (-16/6 - 12/6) = -1/6 - (-28/6) = -1/6 + 28/6 = 27/6 = 9/2 = 4.5

12)

• ∫₄⁶ (3 + x/2) dx = [3x + x²/4]₄⁶ = (3(6) + (6)²/4) - (3(4) + (4)²/4) = (18 + 36/4) - (12 + 16/4) = (18 + 9) - (12 + 4) = 27 - 16 = 11

13)

• ∫₁⁹ 6/√x dx = ∫₁⁹ 6x⁻¹/₂ dx = [12x¹/₂]₁⁹ = 12√(9) - 12√(1) = 12(3) - 12(1) = 36 - 12 = 24

14)

• ∫₋₂⁴ (8 + 2x - x²) dx = [8x + x² - x³/3]₋₂⁴ = (8(4) + (4)² - (4)³/3) - (8(-2) + (-2)² - (-2)³/3) = (32 + 16 - 64/3) - (-16 + 4 + 8/3) = (48 - 64/3) - (-12 + 8/3) = (144/3 - 64/3) - (-36/3 + 8/3) = 80/3 - (-28/3) = 80/3 + 28/3 = 108/3 = 36

15)

• ∫₋₂³ (4x³ - 3x² + 2x + 1) dx = [x⁴ - x³ + x² + x]₋₂³ = ((3)⁴ - (3)³ + (3)² + (3)) - ((-2)⁴ - (-2)³ + (-2)² + (-2)) = (81 - 27 + 9 + 3) - (16 - (-8) + 4 - 2) = (54 + 12) - (16 + 8 + 4 - 2) = 66 - 26 = 40

16)

• ∫₁² (x⁴ + 1) dx = [x⁵/5 + x]₁² = ((2)⁵/5 + (2)) - ((1)⁵/5 + (1)) = (32/5 + 2) - (1/5 + 1) = (32/5 + 10/5) - (1/5 + 5/5) = 42/5 - 6/5 = 36/5 = 7.2

17)

• ∫₀.₅¹ dx/x³ = ∫₀.₅¹ x⁻³ dx = [-1/(2x²)]₀.₅¹ = (-1/(2(1)²)) - (-1/(2(0.5)²)) = (-1/2) - (-1/(2(1/4))) = -1/2 - (-1/(1/2)) = -1/2 - (-2) = -1/2 + 2 = 3/2 = 1.5

18)

• ∫₁⁹ (x-1)/√x dx = ∫₁⁹ (x/√x - 1/√x) dx = ∫₁⁹ (x¹/₂ - x⁻¹/₂) dx = [2/3x³/₂ - 2x¹/₂]₁⁹ = (2/3(9)³/₂ - 2√(9)) - (2/3(1)³/₂ - 2√(1)) = (2/3(27) - 2(3)) - (2/3(1) - 2(1)) = (18 - 6) - (2/3 - 2) = 12 - (2/3 - 6/3) = 12 - (-4/3) = 12 + 4/3 = 36/3 + 4/3 = 40/3 ≈ 13.33

19)

• ∫₁⁸ ³√x² dx = ∫₁⁸ x²/³ dx = [3/5x⁵/³]₁⁸ = 3/5(8)⁵/³ - 3/5(1)⁵/³ = 3/5(32) - 3/5(1) = 96/5 - 3/5 = 93/5 = 18.6