Question

Integrate (x⁴+x²+1)/(x²-x+1). Please solve it in detail....

Answer 1

long division will help as the integral is improper

Answer 2

To find::

$\sf{\int \frac{ {x}^{4} + {x}^{2} + 1 }{ {x}^{2} - x + 1 }dx }$

Add x² and subtract x² in numerator

$\sf{ \int \frac{ ({x}^{4} + 2 {x}^{2} + 1) - {x}^{2} }{ {x}^{2} - x + 1 } dx}$

=>x⁴+2x²+1=(1+x²)²

so

$\sf{ \int \frac{ {(1 + {x}^{2}) }^{2} - {x}^{2} }{ {x}^{2} - x + 1} \: dx }$

we can write a²-b² as (a+b)(a-b)

$\sf{ \int \frac{(1 + {x}^{2} + x)(1 + {x}^{2} - x) }{ {x}^{2} - x + 1 } dx}$

$\sf{ \int(1 + {x}^{2} + x)dx}$

$= > \sf{ \int{1}dx} + \int {x}^{2} dx + \int(x)dx$

$\sf{ = > x + \frac{ {x}^{3} }{3} + \frac{ {x}^{2} }{2} }$