Question

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

Answer 1

`1/2x^2-lnx-x^-2+c`

Explanation:

`"divide the rational function term by term"`

`rArrx^5/x^4-x^3/x^4+(2x)/x^4=x-x^-1+2x^-3`

`"integrate each term using the "color(blue)"power rule"`

`int(ax)^n=a/(n+1)x^(n+1)to(n!=-1)`

`"note that "int(x^-1)=int(1/x)=lnx`

`rArrint(x-x^-1+2x^-3)dx`

`=1/2x^2-lnx-x^-2+c`

`"where c is the constant of integration"`