Question

Ayudennnmeeeeee con proceso por favor :'c

Answer 1

Respuesta:

1) ∫√2x dx / ∛27x = 2√2x⁷/⁶ / 7  + C₂

2) ∫ (x³ + 5x² - 7x - 5) / (x - 3) = x³ / 3 + 4x² + 17x + 46ln(x - 3) + C₅

3) ∫ (x³ - x² - 8x - 9) / (x² + 1) = x² / 2 - x -  9ln(x² + 1) / 2 + 8tan⁻¹x + C₅

Explicación paso a paso:

1)

∫ √2√x dx / ∛27∛x =

∫ √2√x dx / 3∛x =

√2 / 3 ∫ x¹/² dx / x¹/³ =

√2 / 3 ∫ x¹/² ⁻ ¹/³ dx =

√2 / 3 ∫ x⁽³ ⁻ ²⁾/⁶ dx =

√2 / 3 ∫ x¹/⁶ dx =

√2 / 3 [(x⁷/⁶) / (7 / 6) + C₁] =

√2 / 3 [6(x⁷/⁶) / 7  + C₁] =

√2 / 3 [6(x⁷/⁶) / 7]  + √2 / 3C₁ =

2√2x⁷/⁶ / 7  + C₂

2)

(x³ + 5x² - 7x - 5) / (x - 3)

Utilizar método de Ruffini:

       |   1      5    - 7  |  - 5

x = 3|          3    24  |   51

___ |__________ |____

       |   1     8     17  |   46

 

D(x) = x³ + 5x² - 7x - 5

d(x) = x - 3

q(x) = x² + 8x + 17

R(x) = 46  

D(x) = d(x)q(x) + R(x)    

D(x) / d(x) = q(x) + R(x) / d(x)

(x³ + 5x² - 7x - 5) / (x - 3) = (x² + 8x + 17) + 46 / (x - 3)

∫ (x³ + 5x² - 7x - 5) dx / (x - 3) =

∫ (x² + 8x + 17) dx + ∫ 46 dx / (x - 3) =

∫ x² dx + ∫ 8x dx + ∫ 17 dx + 46 ∫ dx / (x - 3) =

∫ x² dx + 8 ∫ x dx + 17 ∫ dx + 46 ∫ dx / (x - 3) =

x³ / 3 + C₁ + 8x² / 2 + C₂ + 17x + C₃ + 46ln(x - 3) + C₄ =

x³ / 3 + 4x² + 17x + 46ln(x - 3) + C₅

3)

(x³ - x² - 8x - 9) / (x² + 0x + 1)

Utilizar método de Horner:

 1  |   1     - 1   |  - 8   - 9

 0 |           0  |  - 1  

- 1  |         - 1   |    0      1

__ |_______|________

    |   1     - 1   |  - 9   - 8

D(x) = x³ - x² - 8x - 9

d(x) = x² + 1

q(x) = x - 1

R(x) = - 9x - 8    

D(x) = d(x)q(x) + R(x)    

D(x) / d(x) = q(x) + R(x) / d(x)  

(x³ - x² - 8x - 9) / (x² + 1) = (x - 1) + (- 9x - 8) / (x² + 1)

(x³ - x² - 8x - 9) / (x² + 1) = (x - 1) - (9x + 8) / (x² + 1)

∫ (x³ - x² - 8x - 9) dx / (x² + 1) =

∫ (x - 1) dx - ∫ (9x + 8) dx / (x² + 1) =

∫ xdx - ∫ dx - ∫ 9x dx / (x² + 1) + ∫ 8dx / (x² + 1) =

∫ xdx - ∫ dx - 9 ∫ x dx / (x² + 1) + 8 ∫ dx / (x² + 1) =

∫ xdx - ∫ dx - (9 / 2) ∫ 2x dx / (x² + 1) + 8 [tan⁻¹x + C₄] =

x² / 2 + C₁ - (x + C₂) -  (9 / 2) [ln(x² + 1) + C₃] + 8tan⁻¹x + 8C₄ =

x² / 2 + C₁ - x - C₂ -  9ln(x² + 1) / 2 - 9C₃/2 + 8tan⁻¹x + 8C₄ =

x² / 2 - x -  9ln(x² + 1) / 2 + 8tan⁻¹x + C₅