# How do you find the antiderivative of (x^5-x^3+2x)/x^4?

May 12, 2018

$\frac{1}{2} {x}^{2} - \ln x - {x}^{-} 2 + c$

#### Explanation:

$\text{divide the rational function term by term}$

$\Rightarrow {x}^{5} / {x}^{4} - {x}^{3} / {x}^{4} + \frac{2 x}{x} ^ 4 = x - {x}^{-} 1 + 2 {x}^{-} 3$

$\text{integrate each term using the "color(blue)"power rule}$

$\int {\left(a x\right)}^{n} = \frac{a}{n + 1} {x}^{n + 1} \to \left(n \ne - 1\right)$

$\text{note that } \int \left({x}^{-} 1\right) = \int \left(\frac{1}{x}\right) = \ln x$

$\Rightarrow \int \left(x - {x}^{-} 1 + 2 {x}^{-} 3\right) \mathrm{dx}$

$= \frac{1}{2} {x}^{2} - \ln x - {x}^{-} 2 + c$

$\text{where c is the constant of integration}$