Question

How do you express `(2x^3 -x^2)/((x^2 +1)^2)` in partial fractions?

Answer 1

Partial fractions are `(2x+1)/(x^2+1)-(2x+1)/(x^2+1)^2`

Explanation:

Let the function `(2x^3-x^2)/(x^2+1)^2` be written in partial fractions as

`(Ax+B)/(x^2+1)+(Cx+D)/(x^2+1)^2`

Solving this becomes `((Ax+B)(x^2+1)+Cx+D)/(x^2+1)^2`

As denominator in given function is same

it follows that

`(Ax+B)(x^2+1)+Cx+DhArr(2x^3-x^2)` or

`Ax^3+Bx^2+(A+C)x+(B+D)hArr(2x^3-x^2)`

Comparing like terms

`A=2, B=-1, A+C=0 and B+D=0` i.e.

`A=2, B=-1, C=-2 and D=-1`

Hence partial fractions are `(2x+1)/(x^2+1)-(2x+1)/(x^2+1)^2`