Question

x³-8/x²-2x,Integrate the given function w.r.t. x considering them well defined and integrable over proper domain.

Answer 1

we have to integrate (x³ - 8)/(x² - 2x)

first of all, we should resolve the function, f(x) = (x³ - 8)/(x² - 2x)

(x³ - 8)/(x² - 2x) = (x³ - 8)/{x(x - 2)}
we know, a³ - b³ = (a - b)(a² + ab + b²)
so, (x³ - 8) = (x³ - 2³) = (x - 2)(x² - 2x + 4)

now, (x - 2)(x² - 2x + 4)/{x(x - 2)}
= (x² - 2x + 4)/x
= x - 2 + 4/x

hence, we have to use (x - 2 + 4/x) in place of (x³ - 8)/(x² - 2x)

so, $\int{(x-2+4/x)}\,dx$

= $\int{x}\,dx-2\int{dx}+4\int{\frac{1}{x}}\,dx$

= $\frac{x^2}{2}-2x+4logx+C$