How do you express (2x^3 -x^2)/((x^2 +1)^2)2x3x2(x2+1)2 in partial fractions?

1 Answer
Feb 14, 2016

Partial fractions are (2x+1)/(x^2+1)-(2x+1)/(x^2+1)^22x+1x2+12x+1(x2+1)2

Explanation:

Let the function (2x^3-x^2)/(x^2+1)^22x3x2(x2+1)2 be written in partial fractions as

(Ax+B)/(x^2+1)+(Cx+D)/(x^2+1)^2Ax+Bx2+1+Cx+D(x2+1)2

Solving this becomes ((Ax+B)(x^2+1)+Cx+D)/(x^2+1)^2(Ax+B)(x2+1)+Cx+D(x2+1)2

As denominator in given function is same

it follows that

(Ax+B)(x^2+1)+Cx+DhArr(2x^3-x^2)(Ax+B)(x2+1)+Cx+D(2x3x2) or

Ax^3+Bx^2+(A+C)x+(B+D)hArr(2x^3-x^2)Ax3+Bx2+(A+C)x+(B+D)(2x3x2)

Comparing like terms

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

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

Hence partial fractions are (2x+1)/(x^2+1)-(2x+1)/(x^2+1)^22x+1x2+12x+1(x2+1)2