Question

How do you integrate `(2x^3-3x^2+x+1)/(-2x+1)`?

Answer 1

`-x^3/3+x^2/2-1/2lnabs(2x-1)+C`

Explanation:

We have to divide polinomials:

`(2x^3-3x^2+x+1):(-2x+1)=-x^2+x`
`-2x^3+x^2`
`----`
`-2x^2+x+1`
`+2x^2-x`
`----`
`1`

`(2x^3-3x^2+x+1)/(-2x+1)=-x^2+x+1/(-2x+1)`
`int(2x^3-3x^2+x+1)/(-2x+1)dx=`
`=int(-x^2+x+1/(-2x+1))dx=`
`=-intx^2dx+intxdx-intdx/(2x-1)=I`
`2x-1=t, 2dx=dt, dx=dt/2`
`I=-x^3/3+x^2/2-intdt/2 1/t=`
`=-x^3/3+x^2/2-1/2lnabs(2x-1)+C`