Question

How do you find the indefinite integral of `int (5x^2 – 4x + 7)/((x^2+1)(2x – 1)) dx`?

Answer 1

`I=-2arctanx+5/2ln|2x-1| +C`

Explanation:

`I=int (5x^2 – 4x + 7)/((x^2+1)(2x – 1)) dx`

`(Ax+B)/(x^2+1)+C/(2x – 1)=((Ax+B)(2x-1)+C(x^2+1))/((x^2+1)(2x – 1)) =`

`=(2Ax^2+2Bx-Ax-B+Cx^2+C)/((x^2+1)(2x – 1)) =`

`=(x^2(2A+C)+x(2B-A)+(-B+C))/((x^2+1)(2x – 1)) = T`

If we compare integrand with expression `T`, we can write:

`2A+C=5`
`2B-A=-4 => A=2B+4`
`-B+C=7 => C=B+7`

`2(2B+4)+B+7=5`
`5B=-10 => B=-2`
`A=0`
`C=5`

`I=int (-2/(x^2+1)+5/(2x-1))dx`

`I=-2arctanx+5/2ln|2x-1| +C`