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Integral de (x ^ 3 - 4x ^ 2 + 6x - 8) / (x * (x ^ 2 - 2x - 8) dx?

Integral de (x ^ 3 - 4x ^ 2 + 6x - 8) / (x * (x ^ 2 - 2x - 8) dx.

Pregunta de Matemáticas · 1 respuesta · 4 votos · mejor respuesta de Makafull

En resumen

<img src="https://tex.z-dn.net/?f=%24%5Cint%5Cfrac%7Bx%5E3-4x%5E2%2B6x-8%7D%7Bx%28x%5E2-2x-8%29%7Ddx%24" /> <img src="https://tex.z-dn.net/?

Respuestas de la comunidad

Mejor respuesta

Makafull
4

<img src="https://tex.z-dn.net/?f=%24%5Cint%5Cfrac%7Bx%5E3-4x%5E2%2B6x-8%7D%7Bx%28x%5E2-2x-8%29%7Ddx%24" />

<img src="https://tex.z-dn.net/?f=%24%5Cfrac%7Bx%5E3-4x%5E2%2B6x-8%7D%7Bx%28x%5E2-2x-8%29%7D%3D1%2B%5Cfrac%7B2%28x%5E2%20-%207x%20%2B%204%29%7D%7Bx%28x%5E2-2x-8%29%7D%24" />

<img src="https://tex.z-dn.net/?f=%24%5Cfrac%7Bx%5E3-4x%5E2%2B6x-8%7D%7Bx%28x%5E2-2x-8%29%7D%3D1%2B%5Cfrac%7B2%28x%5E2%20-%207x%20%2B%204%29%7D%7Bx%28x-4%29%28x%2B2%29%7D%24" />

separemos esta fracción propia

<img src="https://tex.z-dn.net/?f=%24%5Cfrac%7B2%28x%5E2%20-%207x%20%2B%204%29%7D%7Bx%28x-4%29%28x%2B2%29%7D%24" />

<img src="https://tex.z-dn.net/?f=%24%5Cfrac%7B2%28x%5E2%20-%207x%20%2B%204%29%7D%7Bx%28x-4%29%28x%2B2%29%7D%3D%5Cfrac%7BA%7D%7Bx%7D%2B%5Cfrac%7BB%7D%7Bx-4%7D%2B%5Cfrac%7BC%7D%7Bx%2B2%7D%24" />

<img src="https://tex.z-dn.net/?f=x%5E2%28A%20%2B%20B%20%2BC%29%20-%202x%28A%20-%20B%20%2B%202C%29%20-%208A%20%3D%202x%5E2-14x%2B8" />

comparemos coeficientes

<img src="https://tex.z-dn.net/?f=A%2BB%2BC%3D2%5C%5C%20A-B%2B2C%3D7%5C%5CA%3D-1" />

al resolver se obtiene

<img src="https://tex.z-dn.net/?f=A%3D-1%5C%5CB%3D-2%2F3%5C%5CC%3D11%2F3" />

Por ende

<img src="https://tex.z-dn.net/?f=%24%5Cfrac%7Bx%5E3-4x%5E2%2B6x-8%7D%7Bx%28x%5E2-2x-8%29%7D%3D1-%5Cfrac%7B1%7D%7Bx%7D-%5Cfrac%7B2%2F3%7D%7Bx-4%7D%2B%5Cfrac%7B11%2F3%7D%7Bx%2B2%7D%24" />

entonces

<img src="https://tex.z-dn.net/?f=%24%5Cint%20%5Cfrac%7Bx%5E3-4x%5E2%2B6x-8%7D%7Bx%28x%5E2-2x-8%29%7Ddx%3D%5Cint1-%5Cfrac%7B1%7D%7Bx%7D-%5Cfrac%7B2%2F3%7D%7Bx-4%7D%2B%5Cfrac%7B11%2F3%7D%7Bx%2B2%7Ddx%24" />

[img = 10].

Preguntas frecuentes

Integral de (x ^ 3 - 4x ^ 2 + 6x - 8) / (x * (x ^ 2 - 2x - 8) dx?

<img src="https://tex.z-dn.net/?f=%24%5Cint%5Cfrac%7Bx%5E3-4x%5E2%2B6x-8%7D%7Bx%28x%5E2-2x-8%29%7Ddx%24" /> <img src="https://tex.z-dn.net/?