Question

Integrate x^3 - 4x^2 - x dx / x^2 -2x +5​

Answer 1

Step-by-step explanation:

We have,

$\displaystyle \int \dfrac{ {x}^{3} - 4 {x}^{2} - x }{ {x}^{2} - 2x + 5} \: dx$

$\displaystyle = \int \dfrac{x \left( {x}^{2} - 4 x - 1 \right)}{ {x}^{2} - 2x + 5} \: dx$

$\displaystyle = \int \dfrac{x \left( {x}^{2} - 2x + 5 - 2x - 6 \right)}{ {x}^{2} - 2x + 5} \: dx$

$\displaystyle = \int \dfrac{x \left( {x}^{2} - 2x + 5 \right)- 2x ^{2} - 6x }{ {x}^{2} - 2x + 5} \: dx$

$\displaystyle = \int \dfrac{x \left( {x}^{2} - 2x + 5 \right) }{ {x}^{2} - 2x + 5} \: dx - \int \dfrac{2{x}^{2} + 6 }{ {x}^{2} - 2x + 5} \: dx$

$\displaystyle = \int x \: dx - \int \dfrac{2{x}^{2} - 4x + 10 + 4x- 4 }{ {x}^{2} - 2x + 5} \: dx$

$\displaystyle = \int x \: dx - \int \dfrac{2{x}^{2} - 4x + 10 }{ {x}^{2} - 2x + 5} \: dx + \int \dfrac{4x - 4}{{x}^{2} - 2x + 5 } \: dx$

$\displaystyle = \int x \: dx - 2\int \dfrac{{x}^{2} - 2x + 5 }{ {x}^{2} - 2x + 5} \: dx + 2 \int \dfrac{2x - 2}{{x}^{2} - 2x + 5 } \: dx$

$\displaystyle = \int x \: dx - 2\int \: dx + 2 \int \dfrac{d \left( {x}^{2} - 2 x + 5\right)}{{x}^{2} - 2x + 5 }$

$\displaystyle = \dfrac{ {x}^{2} }{2} - 2x + 2 \ln | {x}^{2} - 2x + 5 | + c$