Ayudennnmeeeeee con proceso por favor :'c
Respuestas a la pregunta
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